researcharticle basel iii and the net stable funding...

Hindawi Publishing CorporationISRN Applied MathematicsVolume 2013 Article ID 582707 20 pageshttpdxdoiorg1011552013582707

Research ArticleBasel III and the Net Stable Funding Ratio

F Gideon1 Mark A Petersen2 Janine Mukuddem-Petersen3 and LNP Hlatshwayo3

1 Department of Mathematics Faculty of Science University of Namibia Private Bag 13301 Windhoek 9000 Namibia2 Research Division Faculty of Commerce and Administration North-West University Private Bag x2046Mmabatho 2735 South Africa

3 Economics Division Faculty of Commerce and Administration North-West University Private Bag x2046Mmabatho 2735 South Africa

Correspondence should be addressed to F Gideon fgideonunamna

Received 28 September 2012 Accepted 15 November 2012

Academic Editors S-W Chyuan Y Wang and Y-G Zhao

Copyright copy 2013 F Gideon et alThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We validate the new Basel liquidity standards as encapsulated by the net stable funding ratio in a quantitativemanner In this regardwe consider the dynamics of inverse net stable funding ratio as a measure to quantify the bankrsquos prospects for a stable funding overa period of a year In essence this justifies how Basel III liquidity standards can be effectively implemented in mitigating liquidityproblems We also discuss various classes of available stable funding and required stable funding Furthermore we discuss anoptimal control problem for a continuous-time inverse net stable funding ratio In particular we make optimal choices for theinverse net stable funding targets in order to formulate its cost This is normally done by obtaining analytic solution of the valuefunction Finally we provide a numerical example for the dynamics of the inverse net stable funding ratio to identify trends inwhich banks behavior convey forward looking information on long-term market liquidity developments

1 IntroductionThe episode of financial market turbulence in 2007ndash2009 hasdepicted the importance of liquidity for normal functioningof the financial system It is because of this background thatwe are contributing to the procedures for the regulation andsupervision of sound liquidity risk management for banksSome of the well-documented materials to this regard are thenotable papers by [1ndash4] The Basel Committee on BankingSupervision (BCBS) outlines certain measures to strengthenglobal capital and liquidity regulations The objective forthese measures is to improve the banking sectorrsquos abilityto ensure that risk does not spillover to the real economyThe measures are formulated in a form of a principle forsound liquidity risk management and supervision compris-ing quantitative and qualitative management instruments(see eg [1]) In essence the response provides guidance onrisk management and supervision for funding liquidity riskand promotes a better risk management in that critical areaof financial segment As such the committee will coordinaterigorous followup by supervisors to ensure that banks adhereto these fundamentals principles (see [3] for more details)

The global economic crisis which recently attack the financialsystem occurs due to liquidity constraintsWe define liquidityconstraint as an arbitrary limit on the amount an individualcan borrow or an arbitrary alteration in the interest ratethey pay In some instances banks exchange assets in theform of collateral in order to have access to finances Inessence pledgeable assets support more borrowing whichfurther influence more investment That is the financialfrictions affect corporate investment (see eg [5] for moreinformation) in which tangible bank assets increase theirpotential to access external fundings This is because weexpect tangible assets to mitigate the contractibility problemsin which the value borrowed could be captured by creditorsin default states In the end one would expect that financialfrictions which affect investment decisions in which tangibil-ity has no effect on the cash flow sensitivities of financiallyunconstrained bank

Several investors are facing economic challenges from thebigger market players who have access to overseas financesThe BCBS established two important standards to compli-ment the principle for sound liquidity risk management and

2 ISRN Applied Mathematics

supervision The two standards are expected to achieve somefinancial objectives In the first instance the objective is topromote short-term resilience of a bankrsquos liquidity risk profileby ensuring that the bank has sufficient high-quality liquidassets which is achieved through liquidity coverage ratio(LCR) analysis The LCR is given as a ratio of the value ofthe stock of high-quality liquid assets in stressed conditionsto the total net cash outflows under an observation of the30 days period The purpose of LCR is to ensure that banksmaintain adequate level of high-quality liquid assets in orderto be able to maintain and commit to its obligations In thesecond instance the standards promote the resilience overa longer time horizon by creating additional incentives forbanks to fund their activities with more stable sources offunding on an ongoing basis A protocol of this nature isconducted through net stable funding ratio (NSFR) analysisThis ratio is defined as the available amount of stable fundingto the amount of the required stable funding In this paperwe concentrate on net stable funding ratio (NSFR) whichperforms a complementary role to the LCR by promotingstructural changes in liquidity risk profiles of institutionsaway from short-term funding mismatches and toward morestable longer-term funding of assets and business activitiesIn essence a stable funding is defined as the portion of thosetypes and amounts of equity and liabilities financing expectedto be reliable sources of funds over a one-year time horizonunder conditions of extended stress Therefore an amountof stable funding available comprises liquidity parametersof various types of assets held OBS contingent exposuresincurred andor the activities pursued by the institutionThisstandard is required to be more than 100 to ensure thatthe available funding meet the required funding over theevaluated period This ratio is defined as

Net Stable Funding Ratio (NSFR)

=Available Amount of Stable Funding (ASF)Required Amount of Stable Funding (RSF)

ge 100

(1)

where stable funding includes equity and liability (Tier 1and Tier 2 and stable deposits) reliable over the next yearunder conditions of extended stress and the required amountincludes illiquid assets The main objective of this standard isto provide a framework which the banks employ to repondto the market challenges by ensuring stable fundings onan ongoing basis An additional component of the liquidityframework is a set of monitoring metric to improve cross-border supervisory consistency In Table 4 we represent asummary of the available stable funding (ASF) and therequired stable funding (RSF) components of the net stablefunding ratio (NSFR) together with their multiplicationfactors We define available stable funding (ASF) as the totalamount of a bankrsquos capital preferred stock with maturityequal to one year or more than a year liabilities with effectivematurities of one year or greater demand deposits andorterm deposits with maturities less than a year and wholesalefunding with maturities less than a year In this ratioextended borrowing from the central bank lending facilities

outside regular open market operations is not encouraged inorder not to reliy on the central bank as a source of fundsIn the denominator of the NSFR we have required stablefunding for assets We define RSF as the sum of the assetsheld and funded by the institution multiplied by a specificrequired stable funding factor assigned to each particularasset type This amount is measured using supervisoryassumptions on the broad characteristics of the liquidity riskprofiles of an institutionrsquos assets off-balance sheet exposuresand other selected activities In essence RSF is calculated asthe sum of the value of the assets held and funded by theinstitution multiplied by a specific required stable fundingfactor assigned to each particular asset type added to theamount of OBS activity and multiplied by its associated RSFfactor In Table 5 we enumerate the RSF for the bank over thenext one year in a stress scenario

The overall analysis of the bank liquidity position isconducted through ratio analysis on the bankrsquos balance sheetcomposition In this case the INSFR measures a bankrsquosability to access funding for a 1-year period of acute marketstress In this paper as in Basel III we are interested inthe INSFR that was defined as the sum of interbank assetsand securities issued by public entities as a percentage ofinterbank liabilities The INSFR formula is given by

1

NSFR=

Required Stable Funding (RSF)Available Stable Funding (ASF)

(2)

This ratio of the NSFR standard is designed to

ldquopromote a longer-term funding of the assets andactivities of banking organizations by establishinga minimum acceptable amount of stable fundingbased on the liquidity of an institutionrsquos assets andactivities over a one-year horizonrdquo

The NSFR is a longer-term structural ratio to addressliquidity mismatches and provide incentives for banks to usestable sources to fund their activities The liquidity standardalso outlines a set of standard liquidity monitoring metricsto improve cross-order supervisory consistency These are acommon set of monitoring metrics to assist supervisors inidentifying and analyzing liquidity risk trends at the bankand system-wide level in order to better anticipate risks fromsystemic disruptions

The overall objectivity of the BCBS for establishing thetwo mentioned regimes that is the principles for soundliquidity risk management and regulations and also the twostandards is to ensure that banks maintain high-qualityliquidity levels These levels of bank liquidity determinethe quality of investments opportunities it can entertainin the financial markets and if arbitrageurs (investors)face financial constraints it limits the investment capacitywhich in turn determines the market liquidity (for moreinformation see eg [6]) In most of the cases the economicset up of the global trading market is faced with generalchallenges of liquidity demand and supply Private banksvirtually serve as trading agents and in some instancesthey do demand liquidity with well-articulated investmentopportunities but with no clear future investmentThe public

ISRN Applied Mathematics 3

supply of liquidity affects the private creation of liquidityby banks and the effects interact with firms demand forliquidity to influence investment and capital accumulationWe would say liquidity constrain contributed greatly to thelack of capitalThe formation of capital market imperfectionsconstrains investment during an emerging market financialcrisis and it makes exporters with foreign ownership toincrease capital significantly This is because many bankswith foreign ownership may be able to overcome liquidityconstraints by accessing overseas credit through their parentcompanies In order to strengthen the liquidity manage-ment in the banks our paper devised a stochastic optimalcontrol problem for a continuous-time INSFR model Inessence the control objective is to meet bank INSFR targetsby making optimal choices for the two control variablesthat is the liquidity provisioning rate and asset allocationThis enables us to achieve an analytic solution to thecase of quadratic cost functions (see Theorem 3 for moredetails)

11 Literature Review on BASEL III and Net Stable FundingRatio (NSFR) Reference [7] suggests that the proposedBasel III liquidity standards constitute a cornerstone of theinternational regulatory reaction to the ongoing crisis Inthis subsection we survey the existing literature seeking toaddress liquidity problems experienced during the recentfinancial crisis (see eg [8])The review centres around BaselIII and liquidity are introduced in [9 10]

According to [11] guidance for supervisors has been aug-mented substantially The guidance emphasizes the impor-tance of supervisors assessing the adequacy of a bankrsquosliquidity riskmanagement framework and its level of liquidityand suggests steps that supervisors should take if these aredeemed inadequate This guidance focuses on liquidity riskmanagement at medium and large complex banks but thesoundprinciples have broad applicability to all types of banksThe implementation of the sound principles by both banksand supervisors should be tailored to the size nature ofbusiness and complexity of a bankrsquos activities (see [11] formore details) A bank and its supervisors also should considerthe bankrsquos role in the financial sectors of the jurisdictionsin which it operates and the bankrsquos systemic importancein those financial sectors (see eg [12]) The BCBS fullyexpects banks and national supervisors to implement therevised principles promptly and thoroughly and the BCBSwill actively review progress in implementation During themost severe episode of the crisis themarket lost confidence inthe solvency and liquidity of many banking institutions Theweaknesses in the banking sector were rapidly transmitted tothe rest of the financial system and the real economy resultingin a massive contraction of liquidity and credit availabilityUltimately the public sector had to step in with unprece-dented injections of liquidity capital support and guaranteesexposing taxpayers to large losses In response to this duringFebruary 2008 the BCBS published [11] The difficultiesoutlined in that paper highlighted that many banks had failedto take account of a number of basic principles of liquidityrisk management when liquidity was plentiful Many of themost exposed banks did not have an adequate framework

that satisfactorily accounted for the liquidity risks posedby individual products and business lines and thereforeincentives at the business level were misaligned with theoverall risk tolerance of the bank Many banks had notconsidered the amount of liquidity they might need to satisfycontingent obligations either contractual or noncontractualas they viewed funding of these obligations to be highlyunlikely

According to [13] also in response to the market fail-ures revealed by the crisis the BCBS has introduced anumber of fundamental reforms to the international reg-ulatory framework The reforms strengthen bank level ormicroprudential regulation which will help to raise theresilience of individual banking institutions to periods ofstress Moreover banks are mandated to comply with tworatios that is the LCR and the NSFR to be effectivelyimplemented inmitigating sovereign debt crises (see eg [1415]) The LCR is intended to promote resilience to potentialliquidity disruptions over a thirty day horizon It will helpto ensure that global banks have sufficient unencumberedhigh-quality liquid assets to offset the net cash outflows itcould encounter under an acute short-term stress scenarioThe specified scenario is built upon circumstances experi-enced in the global financial crisis that began in 2007 andentails both institution-specific and systemic shocks (seeeg [12]) On the other hand The NSFR aims to limit over-reliance on the short-term wholesale funding during timesof buoyant market liquidity and encourage better assess-ment of liquidity risk across all on- and off-balance sheetitems

Theminimum tangible common equity requirements willincrease from 2 to 45 in addition banks will be askedto hold a capital conservation buffer of 25 to withstandfuture periods of stress The new liquidity standards willbe focused on two indicators the LCR that imposes tightercontrols on short-term liquidity flows and the NSFR thataims at reducing the maturity mismatch between assetsand liabilities Unfortunately raising capital and liquiditystandards may have a cost Banks may respond to regulatorytightening by passing on additional funding costs to theirretail business by raising lending rates in order to keep thereturn on equity in line with market valuations andor byreducing the supply of credit so as to lower the share of riskyassets in their balance sheets (see eg [16]) Reduced creditavailability and higher financing costs could affect householdandfirm spendingWhile the regulator has taken this concerninto account planning a long transition period (until 2018)the new rules could cause subdued GDP growth in the shortto medium term How large is the effect that the new rulesare likely to have on GDP in Italy Not very according tothe estimates presented in this paper For each percentagepoint increase in the capital requirement implemented overan eight-year horizon the level of GDP relative to the baselinepath would decline at trough by 000ndash033 dependingon the estimation method (003ndash039 including nonspreadeffects) the median decline at trough would be 012 (023including nonspread effects) The trough occurs shortly afterthe end of the transition period thereafter output slowlyrecovers and by the end of 2022 it is above the baseline value


Based on these estimates the reduction of the annual growthrate of output in the transition period would be in a rangeof 000ndash004 percentage points (000ndash005 percentage pointsincluding nonspread effects) The fall in output is drivenfor the most part by the slowdown in capital accumulationwhich suffers from higher borrowing costs (and credit supplyrestrictions) Compliance with the new liquidity standardsalso yields small costs The additional slowdown in annualGDP growth is estimated to be at most 002 percentagepoints (see eg [16]) These results are broadly similar tothose shown in [17] derived for the main G20 economiesIf banks felt forced by competitors or financial marketsto speed up the transition to the new capital standardsthe fall in output could be steeper and quicker Assumingthat the transition is completed by the beginning of 2013GDP would reach a trough in the second half of 2014 foreach percentage point increase in capital requirements GDPwould slow down by 002ndash014 percentage points in each yearof the 2011ndash2013 period (see eg [16]) It would subsequentlyrebound partly compensating the previous fall Long-runcosts of achieving a 1-percentage point increase in the targetcapital ratio are also small (slightly less than 02 of steadystate GDP) those needed to comply with the new liquidityrequirements are of similar size Econometric estimates aretypically subjected to high uncertainty those presented inthis paper are of no exceptionThemain finding of this paperis nonetheless shared by several other studies The economiccosts of stronger capital and liquidity requirements are nothuge and become negligible if compared with the potentialbenefits that can be reaped from reducing the frequency ofsystemic crises and the amplitude of boom-bust cycles [18]evaluates that if the capital ratio increases by 1-percentagepoint relative to the historical average the expected netbenefits in terms of GDP level would be in a range of 020ndash232 (025ndash333 if liquidity requirements are also met)depending on whether financial crises are assumed to havea temporary or a permanent effect on the output Even takingthe most cautious estimate the gains undoubtedly outweighthe costs to be paid to achieve a sounder banking system(see eg [16])

According to [19] the liquidity framework includes amodule to assess risks arising from maturity transformationand rollover risks A liquidity buffer covers small refinancingneeds because of its limited size In the management of risk abank can combine liquidity buffers and transparency to hedgesmall and large refinancing needs A bank that can ldquoproverdquo itssolvency will be able to attract external refinancing

The purpose of the nonstandard monetary policy toolsuggested by Basel III is to ensure that banks build up suf-ficient liquidity buffers on their own to meet cash-flowobligations At the same time the banking system as a wholeaccumulates a stock of liquid reserves to safeguard financialstability Reference [20] suggest that Basel III allows centralbanks to be in charge of overseeing systemic risk which placesthem in a position to focus on system-wide risks Also BaselIII allows for central bank reserves which serve as meanswhereby commercial banks manage their liquidity risk (seeeg [20]) It also assists supervisors to inform the centralbanks of their judgement

12 Main Questions and Article Outline In this subsectionwe pose the main questions and provide an outline of thepaper

121 Main Questions In this paper on bank net stablefunding ratio we answer the following questions

Question 1 (banking model) Can we model banksrsquo requiredstable funding (RSF) and available stable funding (ASF) aswell as inverse net stable funding ratio (INSFR) in a stochasticframework where constraints are considered (compare withSection 3)

Question 2 (bank liquidity in a numerical quantitative frame-work) Can we explain and provide numerical examples ofthe dynamics of bank inverse net stable funding ratio (referto Section 33)

Question 3 (optimal bank control problem) Can we deter-mine the optimal control problem for a continuous-timeinverse net stable funding ratio (refer to Section 4)

122 Article Outline This paper is organized in the followingway The current chapter introduce the paper and presenta brief background of the Basel III and net stable fundingratio In Section 2 we present simple liquidity data in order toprovide insights into the relationship between liquidity andfinancial crises Furthermore Section 3 serves as a generaldescription of the inverse net stable funding ratio modelingwhich is useful to the wider application for solving theoptimal control problem which follows in the subsequentsection In this section we consolidated our results byproviding the dynamic model for inverse net stable fundingratio and it gives a pictorial trend for the bank liquiditydevelopments Section 4 states the optimal bank inverse netstable funding ratio This model is used to formulate theoptimal stochastic INSFR control problem in Section 4 (seeQuestion 3 and Problem 1)This involves the twomain resultsin the paper namelyTheorems 3 and 4The latter theorem issignificant in that it introduces the idea of a reference processto bank INSFR dynamics (compare with Question 3) Finallyin Section 5 we establish the conclusion for the paper and thepossible research future directions

2 Liquidity in Crisis Empirical Evidence

In this subsection we present simple liquidity data in orderto provide insights into the relationship between liquidity andfinancial crises More precisely we discuss bank (see Section21 for more details) and bond (see Section 22) level liquiditydata In both cases we conclude that low liquidity coincidedwith periods of financial crises

21 Bank Level Liquidity Data In this subsection we con-sider quarterly Call Report data on required stable fundingsemirequired stable funding illiquid assets and illiquidguarantees from selected US commercial banks during theperiod fromQ3 2005 to Q4 2008We collected data on 9067


Table 1 Bank level liquidity (in billions)

Quarters Required stable funding Semirequired stable funding Illiquid assets Illiquid guaranteesQ3 2005 80 220 100 170Q4 2005 30 200 200 220Q1 2006 70 220 280 300Q2 2006 150 210 380 490Q3 2006 110 200 410 530Q4 2006 180 230 490 620Q1 2007 190 220 500 710Q2 2007 200 230 600 900Q3 2007 310 215 710 980Q4 2007 270 230 800 1010Q1 2008 250 210 820 1015Q2 2008 240 220 830 1005Q3 2008 260 230 810 995Q4 2008 310 250 780 990

distinct banks yielding 453579 bank-quarter observationsover the aforementioned period

In Table 1 we note that required stable funding decreasedduring the financial crisis after a steady increase beforethis period As liquidity creation procedures (like bailouts)become effective in Q3 2008 and Q4 2008 the vol-ume of required stable funding increased By comparisonthe semirequired stable funding fluctuated throughout theperiod On the other hand the illiquid assets increased untilQ3 2008 and then decreased as bailouts came into effectDuring the financial crisis illiquid guarantee dynamics hada negative correlation with that of required stable funding

22 Bond Level Liquidity Data In this subsection we explorewhether the effect of illiquidity is stronger during times offinancial crisis We present the results for the crisis periodand compare them with those for the period with normalbond market conditions Firstly we provide evidence fromthe descriptive statistics of the key variables for the twosubperiods and then draw our main conclusions basedon this The analysis of the averages of the variables inthese subperiods allows us to gain some important insightsinto the causes of the variation (see [21]) In this regardTable 2 presents the mean and standard deviations for bondcharacteristics (amount issued coupon maturity and age)trading activity variables (traded volume number of tradesand time interval between trades) and liquidity measures(Amihud price dispersion Roll and zero-return measure)The data set consists of more than 23 000US corporatebonds traded over the period from Q3 2005 to Q4 2008

Table 2 clearly supports the view that all liquidity mea-sures indicated lower liquidity levels during the financial cri-sis As an example we consider the average price dispersionmeasure and find that its value is higher during the crisiscompared to the noncrisis period With regard to the tradingactivity variables we find that the average daily volumeand the trade interval at the bondlevel stay approximatelyconstant However the number of trades increases during the

financial crisis These results are consistent with the level ofmarket-wide trading activity wherewe found that during theGFC trading takes place in fewer bonds with an increasednumber of smaller size trades

3 An Inverse Net Stable Funding Ratio Model

In this section we describe aspects of INSFR modelingthat are important for solving the optimal control problemoutlined in Section 4 We show that concepts related toINSFRs such as volatility in available stable funding and bankinvestment returns may be modeled as stochastic processesIn order to construct our NSFR model we take into accountthe results obtained in [22] in a discrete time framework(see also [23]) Here INSFRs are closely linked to availablestable funding This liquidity design caused asymmetric andloss of information opaqueness and intricacy which haddevastating effects on financial markets

31 Description of the Inverse Net Stable Funding Ratio ModelBefore the GFC banks were prosperous with high liquidityprovisioning rates low interest rates and soaring availablestable funding This was followed by the collapse of thehousing market exploding default rates and the effectsthereafterWemake the following assumption to set the spaceand time indexes that we consider in our INSFR model

Assumption 1 (filtered probability space and time indexes)Throughout we assume that we are working with a filteredprobability space (ΩFP) with filtration F

119905119905ge0

on a timeindex set 119879 = [119905

0 1199051]

Furthermore we are able to produce a system of stochas-tic differential equations that provide information aboutrequired stable funding value at time 119905 with 119909

1 Ω times 119879 rarr

R+ denoted by 1199091

119905and appraised available stable funding at

time 119905 with 1199092

Ω times 119879 rarr R+ denoted by 1199092

119905and their

relationship The dynamics of required stable funding 1199091119905


Table 2 Bond level liquidity

Mean Standard deviationNoncrisis Crisis Noncrisis Crisis

Amount issued (bln) 045 054 003 006Coupon () 624 623 005 005Maturity (yr) 775 831 020 018Age (yr) 436 476 012 014Volume (mln) 144 153 022 032Trades 406 533 024 131Trade interval (dy) 338 337 049 048Amihud (bp per mln) 5321 8920 742 3575Price dispersion (bp) 3975 7002 183 2184Roll (bp) 14282 20977 1057 5234Zero return () 002 003 001 001

is stochastic in nature because in part it depends on thestochastic rates of return on bank assets (see [24] for moredetails) as well as cash in- and outflows Also the dynamics ofavailable stable fundings1199092

119905 is stochastic because its value has

a reliance on the rate of change of available stable funding aswell as liquidity provisioning and risk that have randomnessassociated with them Furthermore for 119909 Ω times 119879 rarr R2 weuse the notation 119909

119905to denote

119909119905= [

1199091

119905

1199092

119905

] (3)

and represent the INSFR with 119897 Ω times 119879 rarr R+ by

119897119905=

1199091

119905

1199092

119905

(4)

It is important for banks that 119897119905in (4) has to be sufficiently

high to ensure high INSFRs Obviously low values of 119897119905

indicate that the bank has low liquidity and is at high risk ofcausing a credit crunch Bank liquidity has a heavy relianceon liquidity provisioning rates Roughly speaking this rateshould be reduced for high INSFRs and increased beyond thenormal rate when bank INSFRs are too low In the sequel thestochastic process 1199061 Ω times 119879 rarr R+ is the normal liquidityprovisioning rate per available stable funding unit whose valueat time 119905 is denoted by 119906

1

119905 In this case 1199061

119905119889119905 turns out to be

the liquidity provisioning rate per unit of the available stablefunding over the time period (119905 119905 + 119889119905) A notion related tothis is the bailout rate per unit of the available stable fundingfor higher or lower INSFRs 1199062 Ω times 119879 rarr R+ that willin closed loop be made dependent on the INSFR Here theequity amount is reliant on required stable funding deficitover available stable fundings We denote the sum of 1199061 and1199062 by the liquidity provisioning rate 1199063 Ωtimes119879 rarr R+ that is

1199063

119905= 1199061

119905+ 1199062

119905 forall119905 (5)

Before and during the GFC the INSFR decreased signifi-cantly as a consequence of rising available stable fundingsDuring this period extensive cash outflows took place Banks

bargained on continued growth in the financial markets (seeeg [22]) The following assumption is made in order tomodel the INSFR in a stochastic framework

Assumption 2 (Liquidity Provisioning Rate) The liquidityprovisioning rate 1199063 is predictable with respect to F

119905119905ge0

and provides us with a means of controlling bank INSFRdynamics (see (5) for more details)

The closed loop systemwill be defined such that Assump-tion 2 is met as we will see in the sequelThe dynamics of thechange per unit of the available stable funding 119890 Ωtimes119879 rarr Rare given by

119889119890119905= 119903119890

119905119889119905 + 120590

119890

119905119889119882119890

119905 119890 (119905

0) = 1198900 (6)

where 119890119905is the change per unit of the available stable funding

119903119890 119879 rarr R is the rate of change per unit of the available

stable funding the scalar 120590119890 119879 rarr R is the volatility in thechange per available stable funding unit and119882

119890 Ω times 119879 rarr

R is standard Brownian motion Furthermore we consider

119889ℎ119905= 119903ℎ

119905119889119905 + 120590

ℎ

119905119889119882ℎ

119905 ℎ (119905

0) = ℎ0 (7)

where the stochastic processes ℎ Ω times 119879 rarr R+ are theasset return per unit of required stable funding 119903ℎ rarr R+ isthe rate of required stable funding return per required stablefunding unit the scalar 120590ℎ 119879 rarr R is the volatility in therate of asset returns and 119882

ℎ Ω times 119879 rarr R is standard

Brownian motion Before the GFC risky asset returns weremuch higher than those of riskless assets making the formera more attractive but much riskier investment It is inefficientfor banks to invest all in risky or riskless securities with assetallocation being important In this regard it is necessary tomake the following assumption to distinguish between riskyand riskless assets for future computations

Assumption 3 (bank required stable funding) Suppose fromthe outset that bank required stable funding can be classifiedinto 119899 + 1 asset classes One of these assets is risk free (liketreasury securities) while the assets 1 2 119899 are risky


The risky assets evolve continuously in time and aremodelled using a 119899-dimensional Brownian motion In thismultidimensional context the asset returns in the kth assetclass per unit of the kth class are denoted by 119910

119896

119905 119896 isin N

119899=

0 1 2 119899 where 119910 Ω times 119879 rarr R119899+1 Thus the assetreturn per required stable funding unit may be representedby

119910 = (T (119905) 1199101

119905 119910

119899

119905) (8)

where T(119905)119905 represents the return on riskless assets and1199101

119905 119910

119899

119905represents the risky return Furthermore we can

model 119910 as

119889119910119905= 119903119910

119905119889119905 + Σ

119910

119905119889119882119910

119905 119910 (119905

0) = 1199100 (9)

where 119903119910 119879 rarr R119899+1 denotes the rate of asset returns Σ119910119905isin

R(119899+1)times119899 is a matrix of asset returns and 119882119910 Ω times 119879 rarr R119899

is a standard Brownian Notice that there are only 119899 scalarBrownian motions due to one of the assets being riskless

We assume that the investment strategy 120587 119879 rarr R119899+1 isoutside the simplex

119878 = 120587 isin R119899+1

120587 = (1205870 120587

119899)119879

1205870+ sdot sdot sdot + 120587

119899= 1

1205870ge 0 120587

119899ge 0

(10)

In this case short selling is possible The required stablefunding returns are then ℎ Ωtimes119877 rarr R+ where the dynamicsof ℎ can be written as

119889ℎ119905= 120587119879

119905119889119910119905= 120587119879

119905119903119910

119905119889119905 + 120587

119879

119905Σ119910

119905119889119882119910

119905 (11)

This notation can be simplified as follows We denote that

119903T(119905) = 119903

1199100

(119905) 119903T 119879 997888rarr R

+

the rate of return on riskless assets

119903119910

119905= (119903

T(119905)

119903119910

119905

119879

+ 119903T(119905) 1119899)

119879

119910 119879 997888rarr R

119899

120587119905= (1205870

119905 119879

119905)119879

= (1205870

119905 1205871

119905 120587

119896

119905)119879

119879 997888rarr R119896

Σ119910

119905= (

0 sdot sdot sdot 0

Σ119910

119905

) Σ119910

119905isin R119899times119899

119905= Σ119910

119905

Σ119910

119905

119879

Thenwe have that

120587119879

119905119903119910

119905= 1205870

119905119903T(119905) +

119895119879

119905119910

119905+ 119895119879

119905119903T(119905) 1119899

= 119903T(119905) +

119879

119905119910

119905

120587119879

119905Σ119910

119905119889119882119910

119905= 119879

119905Σ119910

119905119889119882119910

119905

119889ℎ119905= [119903

T(119905) +

119879

119905119910

119905] 119889119905 +

119879

119905Σ119910

119905119889119882119910

119905 ℎ (119905

0) = ℎ0

(12)

Next we take 119894 Ω times 119879 rarr R+ as the available stable fundingincrease before change per unit of available stable funding119903119894 119879 rarr R+ is the rate of increase of available stable fundings

before change per available stable funding unit the scalar120590119894

isin R is the volatility in the increase of available stablefunding before change and 119882

119894 Ω times 119879 rarr R represents

standard Brownian motion Then we set

119889119894119905= 119903119894

119905119889119905 + 120590

119894119889119882119894

119905 119894 (119905

0) = 1198940 (13)

The stochastic process 119894119905in (13) may typically originate

from available stable funding volatility that may result fromchanges in market activity supply and inflation

We can choose from two approaches when modelingour INSFR in a stochastic setting The first is a realisticmodel that incorporates all the aspects of the INSFR likeavailable stable funding required stable fundings and riskslinked with liquidity Alternatively we can develop a simplemodel which acts as a proxy for somethingmore realistic thatemphasizes features that are specific to our particular studyIn our situation we choose the latter option with the modelfor required stable fundings 1199091 and available stable funding1199092 and their relationship being derived as

1198891199091

119905= 1199091

119905119889ℎ119905+ 1199092

1199051199063

119905119889119905 minus 119909

2

119905119889119890119905

= [119903T(119905) 1199091

119905+ 1199091

119905119879

119905119910

119905+ 1199092

1199051199061

119905+ 1199092

1199051199062

119905minus 1199092

119905119903119890

119905] 119889119905

+ [1199091

119905119879

119905Σ119910

119905119889119882119910

119905minus 1199092

119905120590119890119889119882119890

119905]

1198891199092

119905= 1199092

119905119889119894119905minus 1199092

119905119889119890119905

= 1199092

119905[119903119894

119905119889119905 + 120590

119894119889119882119894

119905] minus 1199092

119905[119903119890

119905119889119905 + 120590

119890119889119882119890

119905]

= 1199092

119905[119903119894

119905minus 119903119890

119905] 119889119905 + 119909

2

119905[120590119894119889119882119894

119905minus 120590119890119889119882119890

119905]

(14)

The SDEs (14) may be rewritten into matrix-vector form inthe following way

Definition 1 (stochastic system for the INSFRmodel) Definethe stochastic system for the INSFR model as

119889119909119905= 119860119905119909119905119889119905 + 119873 (119909

119905) 119906119905119889119905 + 119886

119905119889119905 + 119878 (119909

119905 119906119905) 119889119882119905 (15)

with the various terms in this stochastic differential equationbeing

119906119905= [

1199062

119905

119905

] 119906 Ω times 119879 997888rarr R119899+1

119860119905= [

119903T(119905) minus119903

119890

119905

0 119903119894

119905minus 119903119890

119905

]

119873 (119909119905) = [

1199092

1199051199091

119905

119903119910

119905

119879

0 0

] 119886119905= [

1199092

1199051199061

119905

0]

119878 (119909119905 119906119905) = [

1199091

119905119879

119905Σ119910

119905minus1199092

119905120590119890

0

0 minus1199092

119905120590119890

1199092

119905120590119894]

119882119905= [

[

119882119910

119905

119882119890

119905

119882119894

119905

]

]

(16)


where 119882119910

119905 119882119890119905 and 119882

119894

119905are mutually (stochastically) inde-

pendent standard Brownianmotions It is assumed that for all119905 isin 119879 120590119890

119905gt 0 120590119894

119905gt 0 and

119905gt 0 Often the time argument

of the functions 120590119890 and 120590119894 are omitted

We can rewrite (15) as follows

119873(119909119905) 119906119905= [

1199092

119905

0] 1199062

119905+ [

1199091

119905

119903119910

119905

119879

0

] 119905

= [0 1

0 0] 1199091199051199063

119905+

119899

sum

119898=1

[1199091

119905119910119898

119905

0] 119898

119905

= 11986101199091199051199060

119905+

119899

sum

119898=1

[119910119898

1199050

0 0] 119909119905119898

119905

=

119899

sum

119898=0

[119861119898119909119905] 119906119898

119905

119878 (119909119905 119906119905) 119889119882119905= [

[119879

119905119905119905]12

0

0 0

] 1199091199051198891198821

119905

+ [0 minus120590119890

0 minus120590119890] 1199091199051198891198822

119905+ [

0 0

0 120590119894] 1199091199051198891198823

119905

=

3

sum

119895=1

[119872119895119895(119906119905) 119909119905] 119889119882119895119895

119905

(17)

where 119861 and 119872 are only used for notational purposes tosimplify the equations From the stochastic system given by(15) it is clear that 119906 = (119906

2 ) affects only the stochastic

differential equation of 1199091119905but not that of 1199092

119905 In particular

for (15) we have that affects the variance of 1199091119905and the drift

of 1199091119905via the term 119909

1

119905

119903119910

119905

119879

119905 On the other hand 1199062 affects only

the drift of 1199091119905 Then (15) becomes

119889119909119905= 119860119905119909119905119889119905 +

119899

sum

119895=0

[119861119895119909119905] 119906119895

119905119889119905 + 119886

119905119889119905

+

3

sum

119895=1

[119872119895(119906119905) 119909119905] 119889119882119895119895

119905

(18)

32 Description of the Simplified INSFR Model The modelcan be simplified if attention is restricted to the system withthe INSFR as stated earlier denoted in this section by 119909

119905=

1199091

1199051199092

119905

Definition 2 (stochastic model for a simplified INSFR)Define the simplified INSFR system by the SDE

119889119909119905= 119909119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

+119903119910

119905

119879

119905] 119889119905

+ [1199061

119905+ 1199062

119905minus 119903119890

119905minus (120590119890)2

] 119889119905

+ [(120590119890)2

(1 minus 119909119905)2

+ (120590119894)2

1199092

119905+ 1199092

119905119879

119905119905119905]

12

119889119882119905

119909 (1199050) = 1199090

(19)

The model is derived as follows The starting point is thetwo-dimensional SDE for 119909 = (119909

1 1199092)119879 as in (14) Next we

determine

119889(1199092

119905)minus1

= minus(1199092

119905)minus2

1198891199092

119905+

1

22(1199092)minus3

119905119889 lt 119909

2 1199092gt 119905

= [minus(1199092)minus1

119905(119903119894

119905minus 119903119890

119905) + (119909

2

119905)minus1

((120590119890)2

+ (120590119894)2

)] 119889119905

minus (1199092

119905)minus1

[0 minus120590119890

120590119894] 119889119882119905

119889119909119905= 1199091

119905119889(1199092

119905)minus1

+ (1199092

119905)minus1

1198891199091

119905+ 119889 lt 119909

1 (119909

2)minus1

gt 119905

= [119903T(119905) 119909119905minus 119903119890

119905+ 1199061

119905+ 1199062

119905+ 119909119905119910

119905119905

minus119909119905[119903119894

119905minus 119903119890

119905] + 119909119905((120590119890)2

+ (120590119894)2

) minus (120590119890)2

] 119889119905

+ (119909119905119879

119905Σ119910

119905minus 120590119890(1 minus 119909

119905) minus 120590119894119909119905) 119889119882119905

= 119909119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

+119903119910

119905

119879

119905] 119889119905

+ [1199061

119905+ 1199062

119905minus 119903119890

119905minus (120590119890)2

] 119889119905

+ [(120590119890)2

(1 minus 119909119905)2

+ (120590119894)2

(119909119905)2

+(119909119905)2

119879

119905119905119905+ (120590119894)2

(119909119905)2

]

12

119889119882119905

(20)

for a stochastic process119882 Ωtimes119879 rarr Rwhich is a standardBrownian motion Note that in the drift of the SDE (19) theterm

minus119903119890

119905+ 119909119905119903119890

119905= minus119903119890

119905(119909119905minus 1) (21)

appears because it models the effect of depreciation ofboth required stable fundings and available stable fundingSimilarly the term minus(120590

119890)2+ 119909119905(120590119890)2= (120590119890)2(119909119905minus 1) appears

The predictions made by our previously constructedmodel are consistent with the empirical evidence in contri-butions such as [23 25] For instance in much the same wayas we do [23] describes how available stable fundings affectINSFRs On the other hand to the best of our knowledge themodeling related to collateral and INSFR reference processes(see Section 4 for a comprehensive discussion) has not beentested in the literature before One of the main contributionsof the paper is the way the INSFR model is constructedby using stochastic techniques We believe that this is anaddition to the preexisting literature because it capturessome of the uncertainty associated with INSFR variables Inthis regard we provide a theoretical-quantitative modeling


A trajectory for INSFR

Time (years)

Amountofnetstablefunding

0 01 02 03 04 05 06 07 08 09 1

5

0

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

minus45

Figure 1 Trajectory of the INSFR of stable fundings

Table 3 Choices of inverse net stable funding ratio parameters

Parameter Value119862 500119903119894 001119910 0061199062 003

119903119862 005120590119890 16

04 150119903119890 004120590119894 18

1199061 001

119882 003

framework for establishing bank INSFR reference processesand themaking of decisions about liquidity provisioning ratesand asset allocation

33 Liquidity Simulation In this subsection we provide asimulation of the INSFR dynamics given in (19)

331 INSFR Simulation In this subsubsection we provideparameters and values for a numerical simulation Theparameters and their corresponding values for the simulationare shown below

332 INSFR Dynamics Numerical Example In Figure 1 weprovide the INSFR dynamics in the form of a trajectoryderived from (19)

333 Properties of the INSFR Trajectory Numerical ExampleFigure 1 shows the simulated trajectory for the INSFR forstable funding for the bank Here different values of bankingparameters are collected in Table 3 The number of jumpsof the trajectory was limited to 500 with the initial values

for 119868 fixed at 100 The new Basel III formulated quantitativeframework in the formof standards which play some comple-mentary role to the existing principles for sound liquidity riskmanagement and regulations In this framework we intendto regulate liquidity risk in the banks adopting quantitativemethod This typically involves setting a liquidity ratio asa minimum requirement which however complementedby broader systems and control related to management ofliquidity risk

As we know banks manage their liquidity by offsettingliabilities via assets It is actually the diversification of thebankrsquos assets and liabilities that expose them to liquidityshocks Here we use ratio analysis (in the form of the INSFR)to manage liquidity risk relating various components in thebankrsquos balance sheets Figure 1 depicts that the bank had somedifficulties to secure some stable funding between 01 le 119905 le

03 and also 07 le 119905 le 09 The ratio for INSFR dictates thatthe amount of net stable funding should bele1Hencewe havesome higher liquidity ratio between 04 le 119905 le 07 and thisis because the growth of the required stable funding by thebank was higher than that of the available stable funding Atthis stage the bank might have diversified ways of attractingresources through issuing new bonds and selling securities

There was an even sharper increase subsequent to 119905 = 07

which comes as somewhat of a surprise In order to mitigatethe aforementioned increase in liquidity risk banks can useseveral facilities such as repurchase agreements to securemore funding In order for banks to improve liquidity theymay use debt securities that allow savings from nonfinancialprivate sectors a good network of branches and othercompetitive strategies It is important for banks that 119897

119905in (4)

has to be sufficiently high to ensure high INSFRs Obviouslylow values of 119897

119905indicate that the bank has low liquidity and is

at high risk of causing a credit crunchBank liquidity has a heavy reliance on liquidity provi-

sioning rates Roughly speaking this rate should be reducedfor high INSFRs and increased beyond the normal rate whenbank INSFRs are too low

4 Optimal Bank Inverse Net StableFunding Ratios

In order for a bank to determine an optimal bank bailoutrate (seen as an adjustment to the normal provisioning rate)and asset allocation strategy it is imperative that a well-defined objective function with appropriate constraints isconsidered The choice has to be carefully made in order toavoid ambiguous solutions to our stochastic control problem

41 The Optimal Bank INSFR Problem In our contributionwe choose to determine a control law 119892(119905 119909

119905) that minimizes

the cost function 119869 G119860

rarr R+ where G119860is the class of

admissible control lawsG119860= 119892 119879 timesX 997888rarr U|

119892Borel measurable function

and there exists a unique solution

to the closed-loop system

(22)


Table4Summaryof

netstablefun

ding

ratio

Availables

tablefun

ding

(sou

rces)

Requ

iredstablefund

ing(uses)

Item

Avlblty

Fctr

Item

Rqrd

Fctr

(i)T1KandT2

Kinstr

uments

(ii)O

ther

preferredshares

andcapitalinstrum

entsin

excessof

T2Kallowableam

ount

having

aneffectiv

ematurity

of1Y

rorgt

1Yr

(iii)Other

liabilitiesw

ithan

effectiv

ematurity

of1Y

rorgt

1Yr

100

(i)Ca

sh(ii)S

hort-te

rmun

securedactiv

elytraded

instr

uments(lt1Y

r)(iii)Securitiesw

ithexactly

offsetting

reverser

epo

(iv)S

ecurities

with

remaining

maturity

lt1Y

r(v)N

onrenewableloanstofin

ancialsw

ithremaining

maturity

lt1Y

r

0

Stabledepo

sitso

fretailand

smallbusinessc

ustomers

(non

maturity

orresid

ualm

aturity

lt1Y

r)90

Debtissuedor

guaranteed

bysovereignscentralbank

sBISIM

FEC

non

centralgovernm

entandmultilateraldevelopm

entb

anks

with

a0ris

kweightu

nder

BaselIIS

tand

ardizedAp

proach

5

Lessstabledepo

sitso

fretailand

smallbusinessc

ustomers

(non

maturity

orresid

ualm

aturity

lt1Y

r)80

Unencum

beredno

nfinancialsenioru

nsecured

corporateb

onds

and

coveredbo

ndsrated

atleastA

Aminusanddebt

thatisissuedby

SovereignsC

entralBa

nksandPS

Eswith

arisk

weightin

gof

20

maturity

ge1Y

r

20

Who

lesalefund

ingprovided

byno

nfinancialcorpo

ratecusto

mers

sovereigncentralbanksm

ultilateraldevelopm

entb

anksand

PSEs

(non

maturity

orresid

ualm

aturity

lt1Y

r)50

(i)Unencum

beredlistedequitysecuritieso

rnon

financialsenior

unsecuredcorporateb

onds

(orc

overed

bond

s)ratedfro

mA+to

Aminus

maturity

ge1Y

r(ii)G

old

(iii)Lo

anstono

nfinancialcorpo

rateclients

SovereignsC

entral

Bank

sandPS

Eswith

amaturity

lt1yr

50

Allotherliabilitiesa

ndequityno

tincludedabove

0

Unencum

beredresid

entia

lmortgages

ofanymaturity

andother

unencumberedloansexclu

ding

loanstofin

ancialinstitu

tions

with

aremaining

maturity

of1Y

rorgt

1Yrthatw

ould

qualify

forthe

35or

lower

riskweightu

nder

baselIIstand

ardizedapproach

forc

redit

risk

65

Other

loanstoretailclientsandsm

allbusinessesh

avingam

aturity

lt1Y

r85

Allothera

ssets

100

Offbalances

heetexpo

sures

Und

rawnam

ount

ofcommitted

creditandliq

uidityfacilities

5

Other

contingent

fund

ingob

ligations

National

superviso

rydiscretio

n


Table 5 Detailed composition of asset categories and associated RSF factors

RSF factor Components of RSF category(i) Cash immediately available to meet obligations not currently encumbered as collateral and not held for planned use (ascontingent collateral salary payments or for other reasons)

(ii) Unencumbered short-term unsecured instruments and transactions with outstanding maturities of less than one year1

0 (iii) Unencumbered securities with stated remaining maturities of less than one year with no embedded options that wouldincrease the expected maturity to more than one year(iv) Unencumbered securities held where the institution has an offsetting reverse repurchase transaction when the securityon each transaction has the same unique identifier (eg ISIN number or CUSIP)(v) Unencumbered loans to financial entities with effective remaining maturities of less than one year that are not renewableand for which the lender has an irrevocable right to call

5

Unencumbered marketable securities with residual maturities of one year or greater representing claims on or claimsguaranteed by sovereigns central banks BIS IMF EC noncentral government PSEs or multilateral development banks thatare assigned a 0 risk-weight under the Basel II Standardized Approach provided that active repo or sale-markets exist forthese securities

20(i) Unencumbered corporate bonds or covered bonds rated AAminus or higher with residual maturities of one year or greatersatisfying all of the conditions for L2As in the LCR outlined in Paragraph 42(b)(ii) Unencumbered marketable securities with residual maturities of one year or greater representing claims on or claimsguaranteed by sovereigns central banks noncentral government PSEs that are assigned a 20 risk weight under the Basel IIStandardized Approach provided that they meet all of the conditions for Level 2 assets in the LCR outlined in Paragraph42(a)

50

(i) Unencumbered gold(ii) Unencumbered equity securities not issued by financial institutions or their affiliates listed on a recognized exchangeand included in a large cap market index

(iii) Unencumbered corporate bonds and covered bonds that satisfy all of the following conditions

(a) central bank eligibility for intraday liquidity needs and overnight liquidity shortages in relevant jurisdictions2

(b) not issued by financial institutions or their affiliates (except in the case of covered bonds)

(c) not issued by the respective firm itself or its affiliates(d) Low credit risk assets have a credit assessment by a recognized ECAI of A+ to Aminus or do not have a credit assessmentby a recognized ECAI and are internally rated as having a PD corresponding to a credit assessment of A+ to Aminus

(e) traded in large deep and active markets characterized by a low level of concentration(iv) Unencumbered loans to nonfinancial corporate clients sovereigns central banks and PSEs having a remaining maturityof less than one year

65(i) Unencumbered residential mortgages of any maturity that would qualify for the 35 or lower risk weight under Basel IIStandardized Approach for credit risk(ii) Other unencumbered loans excluding loans to financial institutions with a remaining maturity of one year or greaterthat would qualify for the 35 or lower risk weight under Basel II Standardized Approach for credit risk

85 Unencumbered loans to retail customers (ie natural persons) and small business customers (as defined in the LCR) havinga remaining maturity of less than one year (other than those that qualify for the 65 RSF above)

100 All other assets not included in the above categories1Such instruments include but are not limited to short-term government and corporate bills notes and obligations commercial paper negotiable certificatesof deposits reserves with central banks and sale transactions of such funds (eg fed funds sold) bankers acceptances money market mutual funds2See Footnote 8 for further discussion of central bank eligibility

with the closed-loop system for 119892 isin G119860being given by

119889119909119905= 119860119905119909119905119889119905 +

119899

sum

119895=0

119861119895119909119905119892119895(119905 119909119905) 119889119905 + 119886

119905119889119905

+

3

sum

119895=1

119872119895119895(119892 (119905 119909

119905)) 119909119905119889119882119895119895

119905 119909 (119905

0) = 1199090

(23)

Furthermore the cost function 119869 G119860

rarr R+ of the INSFRproblem is given by

119869 (119892) = E [int

1199051

1199050

exp (minus119903119891(119897 minus 1199050)) 119887 (119897 119909

119897 119892 (119897 119909

119897)) 119889119897

+ exp (minus119903119891(1199051minus 1199050)) 1198871(119909 (1199051)) ]

(24)


where 119892 isin G119860 119879 = [119905

0 1199051] and 119887

1 X rarr R+ is a Borel

measurable function Furthermore 119887 119879 timesX timesU rarr R+ isformulated as

119887 (119905 119909 119906) = 1198872(1199062) + 1198873(1199091

1199092) (25)

for 1198872 U2rarr R+ and 119887

3 R+ rarr R+ Also 119903119891 isin R is called

the forecasting rate of available stable funding The functions1198871 1198872 and 119887

3 are selected below where various choices areconsidered In order to clarify the stochastic problem thefollowing assumption should be made

Assumption 4 (admissible class of control laws) Assume thatG119860

= 0

We are now in a position to state the stochastic optimalcontrol problem for a continuous-time INSFR model that wesolve The said problem may be formulated as follows

Problem 1 (optimal bank Insfr problem) Consider thestochastic control system in (23) for the INSFR problem withthe admissible class of control lawsG

119860 given by (22) and the

cost function 119869 G119860

rarr R+ given by (24) Solve

inf119892isinG119860

119869 (119892) (26)

which amounts to determining the value 119869lowast given by

119869lowast= inf119892isinG119860

119869 (119892) (27)

and the optimal control law 119892lowast if it exists

119892lowast= arg min

119892isinG119860

119869 (119892) isin G119860 (28)

42 Optimal Bank INSFRs in the Simplified Case Considerthe simplified system in (19) for the INSFR problem with theadmissible class of control laws G

119860 given by (22) but with

X = R In this section we have to solve


119869 (119892)


119869 (119892)

(29)

119869 (119892) = E [int

1199051

1199050

exp (minus119903119891(119897 minus 1199050)) [1198872(1199062

119905) + 1198873(119909119905)] d119905

+ exp (minus119903119891(1199051minus 1199050)) 1198871(119909 (1199051)) ]

(30)

where 1198871 R rarr R+ 1198872 R rarr R+ and 119887

3 R+ rarr R+

are all Borel measurable functions For the simplified casethe optimal cost function in (29) should be determined withthe simplified cost function 119869(119892) given by (30) In this caseassumptions have to be made in order to find a solution forthe optimal cost function 119869lowast Next we state and prove animportant result

Theorem 3 (optimal bank INSFRS in the simplified case)Suppose that 1198922lowast and 119892

3lowast are the components of the optimalcontrol law 119892lowast that deal with the optimal bailout rate 1199062lowastand optimal asset allocation 120587119896lowast respectively Consider thenonlinear optimal stochastic control problem for the simplifiedINSFR system in (19) formulated in Problem 1 Suppose that thefollowing assumptions hold

Assumption 31 The cost function is assumed to satisfy

1198872(1199062) isin 1198622(R)

lim1199062rarrminusinfin

11986311990621198872(1199062) = minusinfin

lim1199062rarr+infin

11986311990621198872(1199062) = +infin

119863119906211990621198872(1199062) gt 0 forall119906

2isin R

(31)

with the differential operator119863 which is applied in this case tofunction 119887

2

Assumption 32 There exists a function V 119879 times R rarr RV isin 11986212

(119879 timesX) which is a solution of the PDE given by

0 = 119863119905V (119905 119909) +

1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909119905)2

]119863119909119909V (119905 119909)

+ 119909 (119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)119863119909V (119905 119909)

+ [1199061

119905minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ 1199062

119905

lowast

119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050)) 1198872(1199062

119905

lowast

)

+ exp (minus119903119891(119905 minus 1199050)) 1198873(119909) minus

[119863119909V (119905 119909)]

2

2119863119909119909V (119905 119909)

119903119910

119905

119879

minus1

119905119910

119905

(32)

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050)) 1198871(119909) (33)

where 1199062lowast is the unique solution of the equation

0 = 119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))11986311990621198872(1199062

119905) (34)

Then the optimal control law is

1198922lowast

(119905 119909) = 1199062lowast

1198922lowast

119879 timesX rarr R+ (35)

with 1199062lowast

isin U2the unique solution of (34)

lowast= minus

119863119909V (119905 119909)

119909119863119909119909V (119905 119909)

minus1

119905119910

119905

1198923119896lowast

(119905 119909) = min 1max 0 119896lowast 1198923119896lowast

119879 timesX rarr R

(36)


Furthermore the value of the problem is

119869lowast= 119869 (119892

lowast) = E [V (119905 119909

0)] (37)

Proof It will be proven that the minimization in the dynamicprogramming equation can be achieved and that there existsa solution to the dynamic programming equation

Step 1 Recall from optimal stochastic control theory (see eg[26]) that the dynamic programming equation (DPE) for theoptimal control problem for119908 119879timesR rarr R119908 isin 119862

12(119879timesX)

is given by

0 = 119863119905119908 (119905 119909)

+ inf1199062isinR isin[01]

[1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2

+(119909)2119879119905]119863119909119909119908 (119905 119909)

+ [ (119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

+119903119910

119905

119879

) 119909]119863119909119908 (119905 119909)

+ [1199061

119905+ 1199062minus 119903119890

119905minus (120590119890)2

]119863119909119908 (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) [1198872(1199062) + 1198873(119909)] ]

119908 (1199051 119909) = exp (minus119903

119891(1199051minus 1199050)) 1198871(119909)

(38)

Note that this DPE separates additively into terms

(a) depending on 1199062

(b) depending on

(c) not depending on either of these variables

The minimization is therefore decomposed into two

Step 2 The minimizations are calculated as follows with thefunction V

inf1199062isinR

1198672(119905 119909 119906

2)

1198672(119905 119909 119906) = 119906

2119863119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) 1198872(1199062)

11986311990621198672(119905 119909 119906) = 119863

119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050))11986311990621198872(1199062) = 0

(39)

Because of Assumption 31 in the hypothesis of the theorem(39) for all (119905 119909) isin 119879 timesX has a unique solution 119906

2lowast

isin U = R

and

inf1199062isinR

1198672(119905 119909 119906

2) = 119867

2(119905 119909 119906

2lowast

)

= 1199062lowast

119863119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) 1198872(1199062lowast

)

(40)

Define the function 1198922(119905 119909)lowast

= 1199062lowast 1198922lowast 119879 times X rarr

R It follows from Assumption 31 in the hypothesis of thetheorem that 1198922lowast is a Borel measurable function Considerthe minimization problem

infisinR119896

1198673(119905 119909 )

1198673(119905 119909 ) =

1

2[119879

119905119905119905] (119909)2119863119909119909V (119905 119909)

+119903119910

119905

119879

119909119863119909V (119905 119909)

=1

2(119905+ (

119909 [119863119909V (119905 119909)]

(119909)2119863119909119909V (119905 119909)

) minus1

119905119910

119905)

119879

times 119905(119909)2119863119909119909V (119905 119909)

times (119905+ (

119909 [119863119909V (119905 119909)]

(119909)2119863119909119909V (119905 119909)

) minus1

119905119910

119905)

minus1

2(

[119909119863119909V (119905 119909)]

2

(119909)2119863119909119909V (119905 119909)

)119903119910

119905

119879

minus1

119905119910

119905

119905(119909)2119863119909119909V (119905 119909) gt 0 forall119909 isin R 0

(41)

Hence we have that

lowast= minus

119863119909V (119905 119909)

119909119863119909119909V (119905 119909)

minus1

119905119910

119905

1198923119896lowast

(119905 119909) =

119896lowast if

119896

sum

119894=1

119894lowast

isin [0 1]

119896lowast

sum119896

119895=1119895lowast

if119896

sum

119894=1

119894lowast

gt 1

forall119896 isin Z119899

1198673(119905 119909

lowast) = minus

[119863119909V (119905 119909)]

2

2119863119909119909V (119905 119909)

(119910

119905)119879

minus1

119905119910

119905

(42)

If for a 119896 isin Z119899 119896lowast notin [0 1] then the DPE (32) has to bemodified with the difference of the terms obtained with and


without the constraint This is not indicated in the currentpaper

Step 3 We can rewrite (32) by using the infima obtained inStep 2 and the DPE for V The resulting equation is

0 = 119863119905V (119905 119909)

+ inf1199062isinRisin[01]

1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2

+(119909)2(119879119905) ]119863

119909119909V (119905 119909)

+ [119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+(120590119894)2

+119903119910

119905

119879

] 119909

+ [1199061

119905+ 1199062minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) [1198872(1199062) + 1198873(119909)]

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050)) 1198871(119909)

(43)

Thus V whose existence is assumed by Assumption 2 in thehypothesis of the theorem is a solution of the dynamicprogramming equation It follows then from the standardoptimal stochastic control as in [26] that 1198922lowast and 119892

3119896lowast arethe optimal control laws and that the value is given by 119869

lowast=

119869(119892lowast) = E[V(119905

0 1199090)]

In Theorem 3 we assume that the cost function in (30)satisfies constraints represented by (31) Secondly there existsa functions (33) that is a solution to (32) Then the optimalcontrol law is given by (35) and (36) where 1199062lowast is a solutionto (34) Finally the optimal cost function is given by (37) Itis of interest to choose particular cost functions for whichan analytic solution can be obtained for the value functionand for the control laws The following theorem provides theoptimal control laws for a particular choice of cost functions

Theorem 4 (optimal bank INSFRs with quadratic costfunctions) Consider the nonlinear optimal stochastic controlproblem for the simplified INSFR system in (19) formulated inProblem 1 Consider the cost function

119869 (119892) = E [int

1199051

1199050

exp (minus119903119891(119897 minus 1199050))

times [1

21198882((1199062)2

(119897)) +1

21198883(119909119905minus 119897119903)2

] 119889119897

+1

21198881(119909 (1199051) minus 119897119903)2 exp (minus119903

119891(1199051minus 1199050)) ]

(44)

It is assumed that the cost functions satisfy

1198871(119909) =

1

21198881(119909 minus 119897

119903)2

1198881isin (0infin)

1198872(1199062) =

1

21198882(1199062)2


(45)

1198873(119909) =

1

21198883(119909 minus 119897

119903)2

1198883isin (0infin) (46)

119897119903isin R called the reference value of the INSFRDefine the first-order ODE

minus119905= minus

(119902119905)2

1198882

+ 1198883+ 1199021199052 (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+ 119902119905[minus119903119891minus119903119910

119905

119879

minus1

119905119910

119905+ (120590119890)2

+ (120590119894)2

] 1199021199051= 1198881

(47)

minus 119903

119905= minus

1198883(119909119903

119905minus 119897119903)

119902119905

minus 119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

minus [1199061

119905minus 119903119890

119905minus (120590119890)2

] minus (119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

)

minus (120590119894)2

119909119903(1199051) = 119897119903

(48)

minus 119897119905= minus119903119891119897119905+ 1198883(119909119903

119905minus 119897119903)2

minus 119902119905(120590119890)2

(119909119903

119905minus 1)2

minus 119902119905(120590119894)2

(119909119903

119905)2

119897 (1199051) = 0

(49)

The function 119909119903 119879 rarr R will be called the INSFR reference

(process) function Then we have the following(a) There exist solutions to the ordinary differential equa-

tions (47) (48) and (49) Moreover for all 119905 isin 119879119902119905gt 0

(b) The optimal control laws are

1199062lowast

119905= minus

(119909 minus 119909119903

119905) 119902119905

1198882

1198922lowast

(119905 119909) = 1199062lowast

1198922lowast

119879 timesX 997888rarr R+

lowast

119905= minus

(119909 minus 119909119903

119905)

119909minus1

119905119910

119905 1198923lowast

119879 timesX 997888rarr R119896

(50)

1198923119896lowast

(119905 119909) = 119896lowast if 119896lowast isin [0 1]

min 1max 0 119896lowast119905 otherwise


(51)

(c) The value function and the value of the problem are

V (119905 119909) = exp (minus119903119891(1199051minus 1199050)) [

1

2(119909 minus 119909

119903

119905)2

119902119905+

1

2119897119905] (52)

119869lowast= 119869 (119892

lowast) = E [V (119905

0 1199090)] (53)


Proof (a) The ordinary differential equation in (47) is ofRiccati type It follows from for example [27 Theorem 37page 364] that a unique solution to this equation exists

The second statement requires a longer argumentRewrite the Riccati differential equation in (47) in the form

minus119905= minus119887(119902

119905)2

+ 119888 + 2119886119905119902119905 1199021199051= 1198881

119887 =1

1198882 119888 = 119888

3isin (0infin) 119886 119879 997888rarr R

(54)

Transform the equation according to

1199021

1199051= 119902 (119905

1minus 119905) 119886

1

119905= 119886 (119905

1minus 119905) 119902

1 [0 1199051minus 1199050] 997888rarr R

1

119905= minus (119905

1minus 119905) = minus119887(119902

1

119905)2

+ 119888 + 21198861

1199051199021

119905 1199021

1199050= 1198881

0 = 1

119905+ 119887(1199021

119905)2

minus 119888 minus 21198861

119905119902119905 1199021

0= 1198881

(55)

Consider the second Riccati differential equation

minus2

119905= minus119887(119902

2

119905)2

+ 119888 1199022

1199051= 1198881 119887 119888 isin (0infin) (56)

The latter Riccati differential equation is time invariant Theassociated first-order linear system with system matrices(0 radic(119887) radic(119888)) is both controllable and observable It thenfollows from a result for this Riccati differential equation that1199022

119905gt 0 for all 119905 isin 119879 = [0 119905

1minus 1199050]

Next a theorem is used from [28 Lemma 51] in thecomparison of the solutions of the two Riccati differentialequations The lemma states that for all 119905 isin [119905

1minus 1199050] 1199021119905

ge

1199022

119905gt 0 if

119887 isin (0infin) (minus119888 0

0 119887) ge (

minus119888 minus1198861

119905

minus1198861

119905119887

) forall119905 isin [0 1199051minus 1199050]

(57)

The matrix inequality is met if and only if

119887 isin (0infin) (119887 minus 119887) ge 0 (119888 minus 119888) ge 0

(119886119905)2

le (119888 minus 119888) (119887 minus 119887) = (1198883minus 1198883) (

1

1198882minus

1

1198882)

(58)

By assumption all functions in 119886 and hence those of 1198861119905

=

119886(1199051minus 119905) are bounded on the bounded interval [0 119905

1minus 1199050]

Hence there exists a constant 1198884 isin (0infin) such that (1198861119905)2lt 1198884

Then one can determine real numbers 119887 119888 isin (0infin) such that

119888 minus 119888 = 1198883minus 1198883ge 0 119887 minus 119887 =

1

1198882minus

1

1198882ge 0

(119886119905)2

le 1198884le (119888 minus 119888) (119887 minus 119887) = (119888

3minus 1198883) (

1

1198882minus

1

1198882)

(59)

for example by choosing 1198882 1198883 isin (0infin) both very smallThusthe inequalities are satisfied and for all 119905 isin [0 119905

1minus 1199050] 119902(1199051minus

119905) = 1199021

119905ge 1199022

119905gt 0

The ordinary differential equation (48) is linear (see eg[29]) Hence it follows for such differential equations that aunique solution exists

(119887 119888) The cost function 1198872 satisfies the conditions of

Theorem 3 It will be proven that the function (52) is asolution of the partial differential equation (32) and (33) ofTheorem 3

Note first that

V (119905 119909) = exp (minus119903119891(119905 minus 1199050)) [

1

2(119909 minus 119909

119903

119905)2

119902119905+

1

2119897119905]

119863119905V (119905 119909) = minus119903

119891V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))

times [minus (119909 minus 119909119903

119905) 119903

119905119902119905+

1

2(119909 minus 119909

119903

119905)2

119905+

1

2

119897119905]

119863119909V (119905 119909) = exp (minus119903

119891(119905 minus 1199050)) (119909 minus 119909

119903

119905) 119902119905

119863119909119909V (119905 119909) = exp (minus119903

119891(119905 minus 1199050)) 119902119905

(60)

The optimal control laws are calculated according to theformulas of the previous theorem

So

0 = 119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))11986311990621198872(1199062)

= exp (minus119903119891(119905 minus 1199050)) [(119909 minus 119909

119903

119905) 119902119905+ 11988821199062]

1199062lowast

= minus(119909 minus 119909

119903

119905) 119902119905

1198882

lowast= minus

(119909 minus 119909119903

119905) 119902119905

119909119902119905

minus1

119905119910

119905

(61)

From the partial differential equation in (32) and the terminalcondition in (33) it follows that

1199021199051= 1198881 119909

119903(1199051) = 119897119903 119897 (119905

1) = 0

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050))

times [1

2(119909 minus 119909

119903(1199051))2

1199021199051+

1

2119897 (1199051)]

= exp (minus119903119891(1199051minus 1199050))

1

21198881(119909 minus 119897

119903)2

= exp (minus119903119891(1199051minus 1199050)) 1198871(119909)


exp (+119903119891(119905 minus 1199050))

times [119863119905V (119905 119909) +

1

2[(120590119890)2

(1 minus 119909)2

+(120590119894)2

(119909)2]119863119909119909V (119905 119909)

+ 119909 [119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]119863119909V (119905 119909)

+ [1199061

119905minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ 1199062lowast

119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050)) 1198872(1199062lowast

)

+ exp (minus119903119891(119905 minus 1199050)) 1198873(119909)

minus1

2

(119863119909V (119905 119909))

2

119863119909119909V (119905 119909)

119903119910

119905

119879

minus1

119905119910

119905]

= minus119903119891 1

2(119909 minus 119909

119903

119905)2

119902119905minus

1

2119903119891119897119905minus (119909 minus 119909

119903

119905) 119902119905119903

119905

+1

2(119909 minus 119909

119903

119905)2

119905+

1

2

119897119905

+1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2] 119902119905

+ (119909 minus 119909119903

119905) 119902119905[119909 (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+1199061

119905minus 119903119890

119905minus (120590119890)2

]

minus1

2

(119909 minus 119909119903

119905)2

(119902119905)2

1198882

+1

21198883(119909 minus 119897

119903)2

minus1

2(119909 minus 119909

119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

=1

2(119909)2[ minus 119903119891119902119905+ 119905+ (120590119890)2

119902119905+ (120590119894)2

119902119905

+ 2119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

minus(119902119905)2

1198882

+ 1198883minus 119902119905

119903119910

119905

119879

minus1

119905119910

119905]

+ 119909 [ + 119903119891119909119903

119905119902119905minus 119902119905119903

119905minus 119909119903

119905119905minus (120590119890)2

119902119905

minus 119909119903

119905119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ 119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] +119909119903

119905(119902119905)2

1198882

minus1198883119897119903+ 119909119903

119905119902119905

119903119910

119905

119879

minus1

119905119910

119905]

+1

2(119909)0[ minus 119903119891(119909119903

119905)2

119902119905+ 2119909119903

119905119902119905119903

119905+ (119909119903

119905)2

119905minus 119903119891119897119905

+ (120590119890)2

119902119905minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

]

minus(119909119903

119905)2

(119902119905)2

1198882

+ 1198883(119897119903)2

minus(119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905+ 119897119905]

(62)

It will be proven that the terms at the powers of theindeterminate 119909 are zero thus at (119909)2 (119909)1 and (119909)

0 Theterm at (119909)2 is zero due to the differential equation for 119902 Theterm at (119909)1 equals

minus 119902119905119903

119905minus 119909119903

119905119905+ 119903119891119909119903

119905119902119905minus 2(120590119890)2

119902119905

minus 119909119903

119905119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ 119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

]

+119909119903

119905(119902119905)2

1198882

minus 1198883119897119903+ 119909119903

119905119902119905

119903119910

119905

119879

minus1

119905119910

119905

= minus119902119905119903

119905

+ 119909119903

119905[minus

(119902119905)2

1198882

+ 1198883

+ 2119902119905(119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+119902119905((120590119890)2

+ (120590119894)2

minus 119903119891minus119903119910

119905

119879

minus1

119905119910

119905)]

minus 119909119903

119905119902119905[ (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

minus119903119891minus119903119910

119905

119879

minus1

119905119910

119905]

+ 119902 [minus(120590119890)2

+ (1199061

119905minus 119903119890

119905minus (120590119890)2

)] +119909119903

119905(119902119905)2

1198882

minus 1198883119897119903

= 119902119905[minus119903

119905+

1198883(119909119903

119905minus 119897119903)

119902119905

]

+ 119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ (119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

) + (120590119894)2

+ (1199061

119905minus 119903119890

119905minus (120590119890)2

)

= 0

(63)


Similarly the term of (119909)0 equals

119897119905minus 119903119891119897119905minus 119903119891(119909119903

119905)2

119902119905

+ 2119909119903

119905[119902119905119903

119905+ 119909119903

119905119905] minus (119909

119903

119905)2

119905+ (120590119890)2

119902119905

minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] + 1198883(119897119903)2

minus(119909119903

119905)2

(119902119905)2

1198882

minus (119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

= (using the term with (119909)1)

119897119905minus 119903119891119897119905minus 119903119891(119909119903

119905)2

119902119905+ (120590119890)2

119902119905

minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] + 1198883(119897119903)2

minus(119909119903

119905)2

(119902119905)2

1198882

minus (119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

+ 2119903119891(119909119903

119905)2

119902119905minus 2119909119903

119905119902119905(120590119890)2

minus 2(119909119903

119905)2

119902119905[119903

T+ 119903119890minus 119903119894+ (120590119890)2

+ (120590119894)2

]

+ 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] minus 11988831198971199032119909119903

119905

+2(119909119903

119905)2

(119902119905)2

1198882

+ 2(119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

minus(119909119903

119905)2

(119902119905)2

1198882

+ 1198883(119909119903

119905)2

+ (119909119903

119905)2

1199021199052 (119903

T+ 119903119890minus 119903119894+ (120590119890)2

+ (120590119894)2

)

+ (119909119903

119905)2

119902119905(minus119903119891+ (120590119890)2

+ (120590119894)2

minus119903119910

119905

119879

minus1

119905119910

119905)

= 119897119905minus 119903119891119897119905+ 1198883(119909119903

119905minus 119897119903)2

minus (120590119890)2

119902119905(119909119903

119905minus 1)2

minus (120590119894)2

119902119905(119909119903

119905)2

= 0

(64)

It then follows fromTheorem 3 that the indicated control lawsare the optimal ones (see eg [26])

43 Comments on Optimal Bank INSFRs The control objec-tive is to meet bank INSFR targets by making optimalchoices for the two control variables namely the liquidityprovisioning rate and asset allocation The objective to meetthese targets is formulated as a cost on the INSFR 119909 inthe simplified model The first control variable the liquidityprovisioning rate is formulated as a cost on bank bailouts1199062 that embeds liquidity risk As for the mathematical form

for the cost functions we have considered several optionsthat are discussed below Of course one can formulate anycost function The question then is whether the resultingdynamic programming equation can be solved analytically

We have obtained an analytic solution so far only for the caseof quadratic cost functions (see Theorem 4 for more details)

For the cost on the bank bailout the function in (45) isconsidered where the input variable 119906

2 is restricted to theset R+ If 1199062 gt 0 then the banks should acquire additionalrequired stable funding The cost function should be suchthat bank bailouts are maximized hence 119906

2gt 0 should

imply that 1198872(1199062) gt 0 In general it seems best to maximizeliquidity provisioning For Theorem 4 we have selected thecost function 119887

2(1199062) = (12)119888

2(1199062)2 given by (45) This

penalizes both positive and negative values of 1199062 in equal

ways The only reason for doing so is that then an analyticsolution of the value function can be obtained

The reference process 119897119903 may take the value 08 thatis the threshold for negative INSFR The cost on meetingliquidity provisioning will be encoded in a cost on the INSFRIf the INSFR 119909 gt 119897

119903 is strictly larger than a set value 119897119903

then there should be a strictly positive cost If on the otherhand 119909 lt 119897

119903 then there may be a cost though most bankswill be satisfied and not impose a cost We have selected thecost function 119887

3(119909) = (12)119888

3(119909 minus 119897119903)2 in Theorem 4 given by

(46) This is also done to obtain an analytic solution of thevalue function and that case by itself is interesting Anothercost function that we consider is

1198873(119909) = 119888

3[exp (119909 minus 119897

119903) + (119897119903minus 119909) minus 1] (65)

which is strictly convex and asymmetric in 119909 with respectto the value 119897

119903 For this cost function costs with 119909 gt 119897119903 are

penalized higher than those with 119909 lt 119897119903 This seems realistic

Another cost function considered is to keep 1198873(119909) = 0 for

119909 lt 119897119903An interpretation of the control laws given by (50)

follows The bank bailout rate 1199062lowast is proportional to thedifference between the INSFR 119909 and the reference processfor this ratio 119909119903 The proportionality factor is 119902

1199051198882 which

depends on the relative ratio of the cost function on 1199062

and the deviation from the reference ratio (119909 minus 119909119903) The

property that the control law is symmetric in 119909 with respectto the reference process 119909

119903 is a direct consequence of thecost function 119887

119903(119909) = (12)119888

3(119909 minus 119909

119903)2 being symmetric

with respect to (119909 minus 119909119903) The optimal portfolio distribution

is proportional to the relative difference between the INSFRand its reference process (119909 minus 119909

119903

119905)119909 This seems natural

The proportionality factor is minus1

119905119910

119905which represents the

relative rates of asset return multiplied with the inverse ofthe corresponding variances It is surprising that the controllaw has this structure Apparently the optimal control lawis not to liquidate first all required stable funding with thehighest liquidity provisioning rate and then the requiredstable fundings with the next to highest liquidity provisioningrate and so onThe proportion of all required stable fundingsdepend on the relative weighting in

minus1

119905119910

119905and not on the

deviation (119909 minus 119909119903

119905)

The novel structure of the optimal control law is thereference process for the INSFR 119909119903 119879 rarr R The differentialequation for this reference function is given by (48) Thisdifferential equation is new for the area of INSFR control and


therefore deserves a discussion The differential equation hasseveral terms on its right-hand side which will be discussedseparately Consider the term

1199061

119905minus 119903119890

119905minus (120590119890)2

(66)

This represents the difference between primary requiredstable funding change and available stable funding Note that1199061

119905is the normal liquidity provisioning rate and 119903

119890

119905is the

rate of required stable funding increase Note that if [1199061119905minus

119903119890

119905minus (120590119890)2] gt 0 then the reference INSFR function can

be increasing in time due to this inequality so that for 119905 gt

1199051 119909119905lt 119897119903The term 119888

3(119909119903

119905minus119897119903)119902119905models that if the reference

INSFR function is smaller than 119897119903 then the function has to

increase with time The quotient 1198883119902119905is a weighting term

which accounts for the running costs and for the effect of thesolution of the Riccati differential equation The term

119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

] (67)

accounts for two effects The difference 119903119890119905minus 119903119894

119905is the net effect

of rate of asset return 119903119890 and that of the change in availablestable funding The term 119903

T(119905) + (120590

119890)2+ (120590119894)2 is the effect of

the increase in the available stable funding due to the risklessasset and the variance of the risky liquidity provisioningThelast term

(119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

) minus (120590119894)2

(68)

accounts for the effect on required stable fundings andavailable stable funding More information is obtained bystreamlining the ODE for 119909119903 In order to rewrite the differ-ential equation for 119909119903 it is necessary to assume the following

Assumption 5 (Liquidity Parameters) Assume that theparameters of the problem are all time invariant and also that119902 has become constant with value 1199020

Then the differential equation for 119909119903 can be rewritten as

minus119903

119905= minus119896 (119909

119903

119905minus 119898) 119909

119903(1199051) = 119897119903

119896 = (119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2

+ (120590119894)2

)) +1198883

1199020

119898 =

11989711990311988831199020minus (1199061minus 119903119890minus (120590119890)2

) + (120590119890)2

(119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2+ (120590119894)2

)) + 11988831199020

(69)

Because the finite horizon is an artificial phenomenon tomake the optimal stochastic control problem tractable it isof interest to consider the long-term behavior of the INSFRreference trajectory 119909119903 If the values of the parameters aresuch that 119896 gt 0 then the differential equation with theterminal condition is stable If this condition holds thenlim119905darr0

119902119905

= 1199020 and lim

119905darr0119909119903

119905= 119898 where the down arrow

prescribes to start at 1199051and to let 119905 decrease to 0 Depending

on the value of119898 the control law at a time very far away fromthe terminal time becomes then

1199062lowast

119905= minus

(119909119905minus 119898) 119902

0

1198882

= gt 0 if 119909

119905lt 119898

lt 0 if 119909119905gt 119898

120587lowast

119905= minus

(119909119905minus 119898)

119909119905

119895

= gt 0 if 119909

119905lt 119898

lt 0 if 119909119905gt 119898 if 120587lowast lt 0 then set 120587lowast = 0

(70)

The interpretation for the two cases follows below

Case 1 (119909119905

gt 119898) Then the INSFR 119909 is too high Thisis penalized by the cost function hence the control lawprescribes not to invest in risky assets The payback advice isdue to the quadratic cost functionwhichwas selected tomakethe solution analytically tractable An increase in the liquidityprovisioning will increase net cash outflow that in turn willlower the INSFR

Case 2 (119909119905lt 119898) The INSFR 119909 is too low The cost function

penalizes and the control law prescribes to invest more inrisky assets In this case more funds will be available andcredit risk on the balance sheet will decrease Thus highervalued required stable fundings can be issued Also banksshould hold less required stable fundings to decrease availablestable funding which will lead in the long run to higherINSFRs

5 Conclusions and Future Directions

In this paper we provide a framework for liquidity manage-ment in the banks In actual sense we provide a descriptionfor the inverse net stable funding ratio dynamics whichpromote the resilience over a longer time horizon by creatingadditional incentives with more stable funding sourcesAlso the paper makes a clear connection between liquidityand financial crises in a numerical-quantitative frameworks(compare with Section 2) In addition we derive a stochasticmodel for INSFR dynamics that depends mainly on requiredstable funding available stable funding as well as the liquidityprovisioning rate (seeQuestion 3) Furthermore we obtainedan analytic solution to an optimal bank INSFR problemwith a quadratic objective function (refer to Question 3)In principle this solution can assist in managing INSFRsHere liquidity provisioning and bank asset allocation areexpressed in terms of a reference process To our knowledgesuch processes have not been considered for INSFRs beforeFurthermore we also provide a numerical example in orderto describe the interplay between the amount of net stablefunding and liquidity demands Specific open questions thatarise out of the discussion in Section 4 are given below TheINSFR has some limitations regarding the characterizationof banksrsquo liquidity positionsTherefore complementary BaselIII ratios such as the net stable funding ratio (NSFR) shouldbe considered for a more complete analysis This shouldtake the structure of the short-term assets and liabilities


of residual maturities into account Further questions thatrequire investigation include the following

(1) What is the value of the variable119898 compared with thevariable 119897119903 which is used in the running cost and in theterminal cost

(2) Is119898 higher or lower than 119897119903

Note that from the formula of119898 follows that

119898 =


) + (120590119890)2

(119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2+ (120590119894)2

)) + 11988831199020

(71)

An expression for the difference 119898 minus 119897119903 is not obvious and

depends on many parameters of the problem as it shouldThere are several other directions in which the results

obtained in this paper can be possibly extended Theseinclude addressing further risk issues aswell as improvementsin the INSFR modeling procedure In this regard instead ofusing a continuous-time stochastic model in order to solvean optimal bank INSFR problem we would like to constructa more sophisticated model with jump-diffusion processes(see eg [30]) Also a study of information asymmetryduring the GFC should be interesting

We have already made several contributions in supportof the endeavors outlined in the previous paragraph Forinstance our paper [24] deals with issues related to liquidityrisk and the GFC Furthermore we started dealing with jumpdiffusion processes in the book chapter [30] Also the role ofinformation asymmetry in a subprime context is related tothe main hypothesis of the book [22]

Appendices

More About Bank INSFRs

In this section we provide more information about requiredstable funding and available stable funding

A Required Stable Funding

In this subsection we discuss the stock of high-qualityrequired stable funding constituted by cash CB reservesmarketable securities and governmentCB bank debt issued

The first component of stock of high-quality requiredstable funding is cash that is it made up of banknotesand coins According to [3] a CB reserve should be ableto be drawn down in times of stress In this regard localsupervisors should discuss and agree with the relevant CB theextent to which CB reserves should count toward the stock ofrequired stable funding

Marketable securities represent claims on or claims guar-anteed by sovereigns CBs noncentral government publicsector entities (PSEs) the Bank for International Settlements(BIS) the International Monetary Fund (IMF) the EuropeanCommission (EC) or multilateral development banks Thisis conditional on all the following criteria being met Theseclaims are assigned a 0 risk weight under the Basel IIStandardizedApproach Also deep repomarkets should exist

for these securities and that they are not issued by banks orother financial service entities

Another category of stock of high-quality required stablefunding refers to governmentCB bank debt issued in domesticcurrencies by the country in which the liquidity risk is beingtaken by the bankrsquos home country (see eg [3 4])

B Available Stable Funding

Cash outflows are constituted by retail deposits unsecuredwholesale funding secured funding and additional liabilities(see eg [3]) The latter category includes requirementsabout liabilities involving derivative collateral calls related toa downgrade of up to 3 notches market valuation changeson derivatives transactions valuation changes on postednoncash or nonhigh quality sovereign debt collateral secur-ing derivative transactions asset backed commercial paper(ABCP) special investment vehicles (SIVs) conduits andspecial purpose vehicles (SPVs) as well as the currentlyundrawn portion of committed credit and liquidity facilities

Cash inflows are made up of amounts receivable fromretail counterparties amounts receivable from wholesalecounterparties and amounts receivable in respect of repo andreverse repo transactions backed by illiquid assets and securi-ties lendingborrowing transactions where illiquid assets areborrowed as well as other cash inflows

According to [3] net cash inflows is defined as cumulativeexpected cash outflows minus cumulative expected cashinflows arising in the specified stress scenario in the timeperiod under consideration This is the net cumulative liq-uidity mismatch position under the stress scenario measuredat the test horizon Cumulative expected cash outflows arecalculated by multiplying outstanding balances of variouscategories or types of liabilities by assumed percentages thatare expected to roll-off and by multiplying specified draw-down amounts to various off-balance sheet commitmentsCumulative expected cash inflows are calculated by multi-plying amounts receivable by a percentage that reflects theexpected inflow under the stress scenario

References

[1] Basel Committee on Banking Supervision Progress Report onBasel III Implementation Bank for International Settlements(BIS) Basel Switzerland 2011

[2] Basel Committee on Banking Supervision Basel III A GlobalRegulatory Framework for More Resilient Banks and BankingSystemsmdashRevised Version Bank for International Settlements(BIS) Basel Switzerland 2011

[3] Basel Committee on Banking Supervision Basel III Interna-tional Framework for Liquidity Risk Measurement StandardsandMonitoring Bank for International Settlements (BIS) BaselSwitzerland 2010

[4] Basel Committee on Banking Supervision Principles for SoundLiquidity Risk Management and Supervision Bank for Interna-tional Settlements (BIS) Basel Switzerland 2008

[5] H Almeida and M Campello ldquoFinancial constraints assettangibility and corporate investmentrdquo The Review of FinancialStudies vol 20 no 5 pp 1429ndash1460 2007


[6] D Gromb and D Vayonos A Model of Financial MarketLiquidity Based on Intermediary Capital London School ofEconomics CEPR and NBER London UK 2009

[7] S W Schmitz ldquoThe impact of the Basel III liquidity stan-dards on the implementation of monetary policyrdquo 2011 httppapersssrncomsol3paperscfmabstract id=1869810

[8] R M Lastra and G Wood ldquoThe crisis of 2007 till 2009 naturecauses and reactionsrdquo Journal of International Economic Lawvol 13 no 3 pp 531ndash550 2009

[9] Basel Committee on Banking Supervision (BCBS) Consul-tative Document International Framework for Liquidity RiskMeasurement Standards and Monitoring Bank for Interna-tional Settlements (BIS) Publications 2009 httpwwwbisorgpublbcbs165htm

[10] Basel Committee on Banking Supervision (BCBS) Basel IIIInternational Framework for Liquidity Risk Measurement Stan-dards and Monitoring Bank for International Settlements (BIS)Publications 2010 httpwwwbisorgpublbcbs188htm

[11] Basel Committee on Banking Supervision (BCBS) Princi-ples for Sound Liquidity Risk Management and SupervisionBank for International Settlements (BIS) Publications 2008httpwwwbisorgpublbcbs144htm

[12] H Hannoun Towards a Global Financial Stability FrameworkBank for International Settlements 2010 httpwwwbisorgspeechessp100303htm

[13] Basel Committee on Banking Supervision (BCBS)ConsultativeDocument Strengthening the Resilience of the Banking SystemBank for International Settlements (BIS) Publications 2009httpwwwbisorgpublbcbs164htm

[14] G Calice J Chen and J Williams ldquoLiquidity interactions incredit markets an analysis of the eurozone sovereign debt cri-sisrdquo Electronic Journal of Economics In press httppapersssrncomsol3paperscfmab-stractid=1776425

[15] N Isaac ldquoEU bailouts fail to keep European Sovereign debtmarkets afloatrdquo April 2011 httpwwwelliottwavecom

[16] A Locarno The Macroeconomic Impact of Basel III on theItalian Economy Occasional Paper no 88 Questioni di Econo-mia e Finanza 2011 httppapersssrncomsol3paperscfmabstract id=1849870

[17] MAG (Macroeconomic Assessment Group) Assessing theMacroeconomic Impact of the Transition to Stronger Capitaland Liquidity Requirements Final ReportThe Financial StabilityBoard and the BCBS 2010

[18] Basel Committee on Banking Supervision (BCBS) An Assess-ment of the Long-Term Economic Impact of Stronger Capitaland Liquidity Requirements Bank for International Settlements(BIS) Publications 2010 httpwwwbisorgpublbcbs175htm

[19] C Schmieder H Hesse B Neudorfer C Puhr and S WSchmitz ldquoNext generation system-wide liquidity stress testingrdquoIMF Working Paper WP123 Monetary and Capital MarketsDepartment 2012

[20] MOjo ldquoPreparing for Basel IVmdashwhy liquidity risks still presenta challenge to regulators in prudential supervisionrdquo Decem-ber 2010 httppapersssrncomsol3paperscfmabstract id=1729057


[22] M A Petersen M C Senosi and J Mukuddem-PetersenSubprime Mortgage Models Nova New York NY USA 2012

[23] G B Gorton The Panic of 2007 Working Paper no 08-24 Yale ICF 2008 httppapersssrncomsol3paperscfmabstract id=1255362

[24] M A Petersen B De Waal J Mukuddem-Petersen and MP Mulaudzi ldquoSubprime mortgage funding and liquidity riskrdquoQuantitative Finance In press

[25] YDemyanyk andO vanHemert ldquoUnderstanding the subprimemortgage crisisrdquo Review of Financial Studies vol 24 no 6 pp1848ndash1880 2011

[26] D P Bertsekas Dynamic Proggramming and Optimal ControlVolumes I and II Athena Scientific 2007

[27] E D SontagMathematical ControlTheory Deterministic FiniteDimensional Systems vol 6 Springer New York NY USA 2ndedition 1998

[28] W Kratz Quadratic Functionals in Variational Analysis andControlTheory vol 6 ofMathematical Topics Akademie BerlinGermany 1995

[29] E A Coddington and R Carlson Linear Ordinary DifferentialEquations Society for Industrial and Applied Mathematics(SIAM) Philadelphia Pa USA 1997

[30] F Gideon M A Petersen J Mukuddem-Petersen and B DeWaal ldquoBank liquidity and the global financial crisisrdquo Journalof Applied Mathematics vol 2012 Article ID 743656 27 pages2012

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Stochastic AnalysisInternational Journal of


supervision The two standards are expected to achieve somefinancial objectives In the first instance the objective is topromote short-term resilience of a bankrsquos liquidity risk profileby ensuring that the bank has sufficient high-quality liquidassets which is achieved through liquidity coverage ratio(LCR) analysis The LCR is given as a ratio of the value ofthe stock of high-quality liquid assets in stressed conditionsto the total net cash outflows under an observation of the30 days period The purpose of LCR is to ensure that banksmaintain adequate level of high-quality liquid assets in orderto be able to maintain and commit to its obligations In thesecond instance the standards promote the resilience overa longer time horizon by creating additional incentives forbanks to fund their activities with more stable sources offunding on an ongoing basis A protocol of this nature isconducted through net stable funding ratio (NSFR) analysisThis ratio is defined as the available amount of stable fundingto the amount of the required stable funding In this paperwe concentrate on net stable funding ratio (NSFR) whichperforms a complementary role to the LCR by promotingstructural changes in liquidity risk profiles of institutionsaway from short-term funding mismatches and toward morestable longer-term funding of assets and business activitiesIn essence a stable funding is defined as the portion of thosetypes and amounts of equity and liabilities financing expectedto be reliable sources of funds over a one-year time horizonunder conditions of extended stress Therefore an amountof stable funding available comprises liquidity parametersof various types of assets held OBS contingent exposuresincurred andor the activities pursued by the institutionThisstandard is required to be more than 100 to ensure thatthe available funding meet the required funding over theevaluated period This ratio is defined as

Net Stable Funding Ratio (NSFR)

=Available Amount of Stable Funding (ASF)Required Amount of Stable Funding (RSF)

ge 100

(1)

where stable funding includes equity and liability (Tier 1and Tier 2 and stable deposits) reliable over the next yearunder conditions of extended stress and the required amountincludes illiquid assets The main objective of this standard isto provide a framework which the banks employ to repondto the market challenges by ensuring stable fundings onan ongoing basis An additional component of the liquidityframework is a set of monitoring metric to improve cross-border supervisory consistency In Table 4 we represent asummary of the available stable funding (ASF) and therequired stable funding (RSF) components of the net stablefunding ratio (NSFR) together with their multiplicationfactors We define available stable funding (ASF) as the totalamount of a bankrsquos capital preferred stock with maturityequal to one year or more than a year liabilities with effectivematurities of one year or greater demand deposits andorterm deposits with maturities less than a year and wholesalefunding with maturities less than a year In this ratioextended borrowing from the central bank lending facilities

outside regular open market operations is not encouraged inorder not to reliy on the central bank as a source of fundsIn the denominator of the NSFR we have required stablefunding for assets We define RSF as the sum of the assetsheld and funded by the institution multiplied by a specificrequired stable funding factor assigned to each particularasset type This amount is measured using supervisoryassumptions on the broad characteristics of the liquidity riskprofiles of an institutionrsquos assets off-balance sheet exposuresand other selected activities In essence RSF is calculated asthe sum of the value of the assets held and funded by theinstitution multiplied by a specific required stable fundingfactor assigned to each particular asset type added to theamount of OBS activity and multiplied by its associated RSFfactor In Table 5 we enumerate the RSF for the bank over thenext one year in a stress scenario

The overall analysis of the bank liquidity position isconducted through ratio analysis on the bankrsquos balance sheetcomposition In this case the INSFR measures a bankrsquosability to access funding for a 1-year period of acute marketstress In this paper as in Basel III we are interested inthe INSFR that was defined as the sum of interbank assetsand securities issued by public entities as a percentage ofinterbank liabilities The INSFR formula is given by

1

NSFR=

Required Stable Funding (RSF)Available Stable Funding (ASF)

(2)

This ratio of the NSFR standard is designed to

ldquopromote a longer-term funding of the assets andactivities of banking organizations by establishinga minimum acceptable amount of stable fundingbased on the liquidity of an institutionrsquos assets andactivities over a one-year horizonrdquo

The NSFR is a longer-term structural ratio to addressliquidity mismatches and provide incentives for banks to usestable sources to fund their activities The liquidity standardalso outlines a set of standard liquidity monitoring metricsto improve cross-order supervisory consistency These are acommon set of monitoring metrics to assist supervisors inidentifying and analyzing liquidity risk trends at the bankand system-wide level in order to better anticipate risks fromsystemic disruptions

The overall objectivity of the BCBS for establishing thetwo mentioned regimes that is the principles for soundliquidity risk management and regulations and also the twostandards is to ensure that banks maintain high-qualityliquidity levels These levels of bank liquidity determinethe quality of investments opportunities it can entertainin the financial markets and if arbitrageurs (investors)face financial constraints it limits the investment capacitywhich in turn determines the market liquidity (for moreinformation see eg [6]) In most of the cases the economicset up of the global trading market is faced with generalchallenges of liquidity demand and supply Private banksvirtually serve as trading agents and in some instancesthey do demand liquidity with well-articulated investmentopportunities but with no clear future investmentThe public

































119905119905ge0


0 1199051]





time 119905 with 1199092


119905and their









119905to denote

119909119905= [

1199091

119905

1199092

119905

] (3)


119897119905=

1199091

119905

1199092

119905

(4)




1

119905 In this case 1199061

119905119889119905 turns out to be


1199063

119905= 1199061

119905+ 1199062

119905 forall119905 (5)




119905119905ge0



119889119890119905= 119903119890

119905119889119905 + 120590

119890

119905119889119882119890

119905 119890 (119905

0) = 1198900 (6)




119890 Ω times 119879 rarr


119889ℎ119905= 119903ℎ

119905119889119905 + 120590

ℎ

119905119889119882ℎ

119905 ℎ (119905

0) = ℎ0 (7)







119896

119905 119896 isin N

119899=


119910 = (T (119905) 1199101

119905 119910

119899

119905) (8)


119905 119910

119899


model 119910 as

119889119910119905= 119903119910

119905119889119905 + Σ

119910

119905119889119882119910

119905 119910 (119905

0) = 1199100 (9)





119878 = 120587 isin R119899+1

120587 = (1205870 120587

119899)119879

1205870+ sdot sdot sdot + 120587

119899= 1

1205870ge 0 120587

119899ge 0

(10)


119889ℎ119905= 120587119879

119905119889119910119905= 120587119879

119905119903119910

119905119889119905 + 120587

119879

119905Σ119910

119905119889119882119910

119905 (11)


119903T(119905) = 119903

1199100

(119905) 119903T 119879 997888rarr R

+


119903119910

119905= (119903

T(119905)

119903119910

119905

119879

+ 119903T(119905) 1119899)

119879

119910 119879 997888rarr R

119899

120587119905= (1205870

119905 119879

119905)119879

= (1205870

119905 1205871

119905 120587

119896

119905)119879

119879 997888rarr R119896

Σ119910

119905= (

0 sdot sdot sdot 0

Σ119910

119905

) Σ119910

119905isin R119899times119899

119905= Σ119910

119905

Σ119910

119905

119879

Thenwe have that

120587119879

119905119903119910

119905= 1205870

119905119903T(119905) +

119895119879

119905119910

119905+ 119895119879

119905119903T(119905) 1119899

= 119903T(119905) +

119879

119905119910

119905

120587119879

119905Σ119910

119905119889119882119910

119905= 119879

119905Σ119910

119905119889119882119910

119905

119889ℎ119905= [119903

T(119905) +

119879

119905119910

119905] 119889119905 +

119879

119905Σ119910

119905119889119882119910

119905 ℎ (119905

0) = ℎ0

(12)






119889119894119905= 119903119894

119905119889119905 + 120590

119894119889119882119894

119905 119894 (119905

0) = 1198940 (13)




1198891199091

119905= 1199091

119905119889ℎ119905+ 1199092

1199051199063

119905119889119905 minus 119909

2

119905119889119890119905

= [119903T(119905) 1199091

119905+ 1199091

119905119879

119905119910

119905+ 1199092

1199051199061

119905+ 1199092

1199051199062

119905minus 1199092

119905119903119890

119905] 119889119905

+ [1199091

119905119879

119905Σ119910

119905119889119882119910

119905minus 1199092

119905120590119890119889119882119890

119905]

1198891199092

119905= 1199092

119905119889119894119905minus 1199092

119905119889119890119905

= 1199092

119905[119903119894

119905119889119905 + 120590

119894119889119882119894

119905] minus 1199092

119905[119903119890

119905119889119905 + 120590

119890119889119882119890

119905]

= 1199092

119905[119903119894

119905minus 119903119890

119905] 119889119905 + 119909

2

119905[120590119894119889119882119894

119905minus 120590119890119889119882119890

119905]

(14)



119889119909119905= 119860119905119909119905119889119905 + 119873 (119909

119905) 119906119905119889119905 + 119886

119905119889119905 + 119878 (119909

119905 119906119905) 119889119882119905 (15)


119906119905= [

1199062

119905

119905

] 119906 Ω times 119879 997888rarr R119899+1

119860119905= [

119903T(119905) minus119903

119890

119905

0 119903119894

119905minus 119903119890

119905

]

119873 (119909119905) = [

1199092

1199051199091

119905

119903119910

119905

119879

0 0

] 119886119905= [

1199092

1199051199061

119905

0]

119878 (119909119905 119906119905) = [

1199091

119905119879

119905Σ119910

119905minus1199092

119905120590119890

0

0 minus1199092

119905120590119890

1199092

119905120590119894]

119882119905= [

[

119882119910

119905

119882119890

119905

119882119894

119905

]

]

(16)


where 119882119910

119905 119882119890119905 and 119882

119894



119905gt 0 120590119894

119905gt 0 and




119873(119909119905) 119906119905= [

1199092

119905

0] 1199062

119905+ [

1199091

119905

119903119910

119905

119879

0

] 119905

= [0 1

0 0] 1199091199051199063

119905+

119899

sum

119898=1

[1199091

119905119910119898

119905

0] 119898

119905

= 11986101199091199051199060

119905+

119899

sum

119898=1

[119910119898

1199050

0 0] 119909119905119898

119905

=

119899

sum

119898=0

[119861119898119909119905] 119906119898

119905

119878 (119909119905 119906119905) 119889119882119905= [

[119879

119905119905119905]12

0

0 0

] 1199091199051198891198821

119905

+ [0 minus120590119890

0 minus120590119890] 1199091199051198891198822

119905+ [

0 0

0 120590119894] 1199091199051198891198823

119905

=

3

sum

119895=1

[119872119895119895(119906119905) 119909119905] 119889119882119895119895

119905

(17)






of 1199091119905via the term 119909

1

119905

119903119910

119905

119879



119889119909119905= 119860119905119909119905119889119905 +

119899

sum

119895=0

[119861119895119909119905] 119906119895

119905119889119905 + 119886

119905119889119905

+

3

sum

119895=1

[119872119895(119906119905) 119909119905] 119889119882119895119895

119905

(18)


119905=

1199091

1199051199092

119905


119889119909119905= 119909119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

+119903119910

119905

119879

119905] 119889119905

+ [1199061

119905+ 1199062

119905minus 119903119890

119905minus (120590119890)2

] 119889119905

+ [(120590119890)2

(1 minus 119909119905)2

+ (120590119894)2

1199092

119905+ 1199092

119905119879

119905119905119905]

12

119889119882119905

119909 (1199050) = 1199090

(19)


1 1199092)119879 as in (14) Next we

determine

119889(1199092

119905)minus1

= minus(1199092

119905)minus2

1198891199092

119905+

1

22(1199092)minus3

119905119889 lt 119909

2 1199092gt 119905


119905(119903119894

119905minus 119903119890

119905) + (119909

2

119905)minus1

((120590119890)2

+ (120590119894)2

)] 119889119905

minus (1199092

119905)minus1

[0 minus120590119890

120590119894] 119889119882119905

119889119909119905= 1199091

119905119889(1199092

119905)minus1

+ (1199092

119905)minus1

1198891199091

119905+ 119889 lt 119909

1 (119909

2)minus1

gt 119905

= [119903T(119905) 119909119905minus 119903119890

119905+ 1199061

119905+ 1199062

119905+ 119909119905119910

119905119905

minus119909119905[119903119894

119905minus 119903119890

119905] + 119909119905((120590119890)2

+ (120590119894)2

) minus (120590119890)2

] 119889119905

+ (119909119905119879

119905Σ119910

119905minus 120590119890(1 minus 119909

119905) minus 120590119894119909119905) 119889119882119905

= 119909119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

+119903119910

119905

119879

119905] 119889119905

+ [1199061

119905+ 1199062

119905minus 119903119890

119905minus (120590119890)2

] 119889119905

+ [(120590119890)2

(1 minus 119909119905)2

+ (120590119894)2

(119909119905)2

+(119909119905)2

119879

119905119905119905+ (120590119894)2

(119909119905)2

]

12

119889119882119905

(20)


minus119903119890

119905+ 119909119905119903119890

119905= minus119903119890

119905(119909119905minus 1) (21)


119890)2+ 119909119905(120590119890)2= (120590119890)2(119909119905minus 1) appears




Time (years)


0 01 02 03 04 05 06 07 08 09 1

5

0

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

minus45



Parameter Value119862 500119903119894 001119910 0061199062 003

119903119862 005120590119890 16

04 150119903119890 004120590119894 18

1199061 001

119882 003











119905in (4)















(22)


Table4Summaryof

netstablefun

ding

ratio

Availables

tablefun

ding

(sou

rces)

Requ

iredstablefund

ing(uses)

Item

Avlblty

Fctr

Item

Rqrd

Fctr

(i)T1KandT2

Kinstr

uments

(ii)O

ther

preferredshares

andcapitalinstrum

entsin

excessof

T2Kallowableam

ount

having

aneffectiv

ematurity

of1Y

rorgt

1Yr

(iii)Other

liabilitiesw

ithan

effectiv

ematurity

of1Y

rorgt

1Yr

100

(i)Ca

sh(ii)S

hort-te

rmun

securedactiv

elytraded

instr

uments(lt1Y

r)(iii)Securitiesw

ithexactly

offsetting

reverser

epo

(iv)S

ecurities

with

remaining

maturity

lt1Y

r(v)N


ancialsw

ithremaining

maturity

lt1Y

r

0

Stabledepo

sitso

fretailand

smallbusinessc

ustomers

(non

maturity

orresid

ualm

aturity

lt1Y

r)90

Debtissuedor

guaranteed


sBISIM

FEC

non

centralgovernm


entb

anks

with

a0ris

kweightu

nder

BaselIIS

tand

ardizedAp

proach

5

Lessstabledepo

sitso

fretailand

smallbusinessc

ustomers

(non

maturity

orresid

ualm

aturity

lt1Y

r)80

Unencum

beredno

nfinancialsenioru

nsecured

corporateb

onds

and

coveredbo

ndsrated

atleastA

Aminusanddebt

thatisissuedby

SovereignsC

entralBa

nksandPS

Eswith

arisk

weightin

gof

20

maturity

ge1Y

r

20

Who

lesalefund

ingprovided

byno

nfinancialcorpo

ratecusto

mers


ultilateraldevelopm

entb

anksand

PSEs

(non

maturity

orresid

ualm

aturity

lt1Y

r)50

(i)Unencum


rnon

financialsenior

unsecuredcorporateb

onds

(orc

overed

bond

s)ratedfro

mA+to

Aminus

maturity

ge1Y

r(ii)G

old

(iii)Lo

anstono

nfinancialcorpo

rateclients

SovereignsC

entral

Bank

sandPS

Eswith

amaturity

lt1yr

50


ndequityno

tincludedabove

0

Unencum

beredresid

entia

lmortgages

ofanymaturity

andother


ding

loanstofin

ancialinstitu

tions

with

aremaining

maturity

of1Y

rorgt

1Yrthatw

ould

qualify

forthe

35or

lower

riskweightu

nder

baselIIstand

ardizedapproach

forc

redit

risk

65

Other


allbusinessesh

avingam

aturity

lt1Y

r85

Allothera

ssets

100

Offbalances

heetexpo

sures

Und

rawnam

ount

ofcommitted

creditandliq

uidityfacilities

5

Other

contingent

fund

ingob

ligations

National

superviso

rydiscretio

n






5



50











119889119909119905= 119860119905119909119905119889119905 +

119899

sum

119895=0

119861119895119909119905119892119895(119905 119909119905) 119889119905 + 119886

119905119889119905

+

3

sum

119895=1

119872119895119895(119892 (119905 119909

119905)) 119909119905119889119882119895119895

119905 119909 (119905

0) = 1199090

(23)



119869 (119892) = E [int

1199051

1199050

exp (minus119903119891(119897 minus 1199050)) 119887 (119897 119909

119897 119892 (119897 119909

119897)) 119889119897

+ exp (minus119903119891(1199051minus 1199050)) 1198871(119909 (1199051)) ]

(24)


where 119892 isin G119860 119879 = [119905

0 1199051] and 119887



119887 (119905 119909 119906) = 1198872(1199062) + 1198873(1199091

1199092) (25)

for 1198872 U2rarr R+ and 119887





= 0







119869 (119892) (26)



119869 (119892) (27)



119892isinG119860

119869 (119892) isin G119860 (28)





119869 (119892)


119869 (119892)

(29)

119869 (119892) = E [int

1199051

1199050

exp (minus119903119891(119897 minus 1199050)) [1198872(1199062

119905) + 1198873(119909119905)] d119905

+ exp (minus119903119891(1199051minus 1199050)) 1198871(119909 (1199051)) ]

(30)


3 R+ rarr R+





1198872(1199062) isin 1198622(R)


11986311990621198872(1199062) = minusinfin


11986311990621198872(1199062) = +infin

119863119906211990621198872(1199062) gt 0 forall119906

2isin R

(31)


2



0 = 119863119905V (119905 119909) +

1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909119905)2

]119863119909119909V (119905 119909)

+ 119909 (119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)119863119909V (119905 119909)

+ [1199061

119905minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ 1199062

119905

lowast

119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050)) 1198872(1199062

119905

lowast

)


[119863119909V (119905 119909)]

2

2119863119909119909V (119905 119909)

119903119910

119905

119879

minus1

119905119910

119905

(32)

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050)) 1198871(119909) (33)


0 = 119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))11986311990621198872(1199062

119905) (34)


1198922lowast

(119905 119909) = 1199062lowast

1198922lowast


with 1199062lowast


lowast= minus

119863119909V (119905 119909)

119909119863119909119909V (119905 119909)

minus1

119905119910

119905

1198923119896lowast



(36)



119869lowast= 119869 (119892

lowast) = E [V (119905 119909

0)] (37)



12(119879timesX)

is given by

0 = 119863119905119908 (119905 119909)


[1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2

+(119909)2119879119905]119863119909119909119908 (119905 119909)

+ [ (119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

+119903119910

119905

119879

) 119909]119863119909119908 (119905 119909)

+ [1199061

119905+ 1199062minus 119903119890

119905minus (120590119890)2

]119863119909119908 (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) [1198872(1199062) + 1198873(119909)] ]

119908 (1199051 119909) = exp (minus119903

119891(1199051minus 1199050)) 1198871(119909)

(38)



(b) depending on




inf1199062isinR

1198672(119905 119909 119906

2)

1198672(119905 119909 119906) = 119906

2119863119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) 1198872(1199062)

11986311990621198672(119905 119909 119906) = 119863

119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050))11986311990621198872(1199062) = 0

(39)


2lowast

isin U = R

and

inf1199062isinR

1198672(119905 119909 119906

2) = 119867

2(119905 119909 119906

2lowast

)

= 1199062lowast

119863119909V (119905 119909)


)

(40)




infisinR119896

1198673(119905 119909 )

1198673(119905 119909 ) =

1

2[119879

119905119905119905] (119909)2119863119909119909V (119905 119909)

+119903119910

119905

119879

119909119863119909V (119905 119909)

=1

2(119905+ (

119909 [119863119909V (119905 119909)]

(119909)2119863119909119909V (119905 119909)

) minus1

119905119910

119905)

119879

times 119905(119909)2119863119909119909V (119905 119909)

times (119905+ (

119909 [119863119909V (119905 119909)]

(119909)2119863119909119909V (119905 119909)

) minus1

119905119910

119905)

minus1

2(

[119909119863119909V (119905 119909)]

2

(119909)2119863119909119909V (119905 119909)

)119903119910

119905

119879

minus1

119905119910

119905

119905(119909)2119863119909119909V (119905 119909) gt 0 forall119909 isin R 0

(41)

Hence we have that

lowast= minus

119863119909V (119905 119909)

119909119863119909119909V (119905 119909)

minus1

119905119910

119905

1198923119896lowast

(119905 119909) =

119896lowast if

119896

sum

119894=1

119894lowast

isin [0 1]

119896lowast

sum119896

119895=1119895lowast

if119896

sum

119894=1

119894lowast

gt 1


1198673(119905 119909

lowast) = minus

[119863119909V (119905 119909)]

2

2119863119909119909V (119905 119909)

(119910

119905)119879

minus1

119905119910

119905

(42)





0 = 119863119905V (119905 119909)


1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2

+(119909)2(119879119905) ]119863

119909119909V (119905 119909)

+ [119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+(120590119894)2

+119903119910

119905

119879

] 119909

+ [1199061

119905+ 1199062minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) [1198872(1199062) + 1198873(119909)]

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050)) 1198871(119909)

(43)



lowast=

119869(119892lowast) = E[V(119905

0 1199090)]



119869 (119892) = E [int

1199051

1199050

exp (minus119903119891(119897 minus 1199050))

times [1

21198882((1199062)2

(119897)) +1

21198883(119909119905minus 119897119903)2

] 119889119897

+1

21198881(119909 (1199051) minus 119897119903)2 exp (minus119903

119891(1199051minus 1199050)) ]

(44)


1198871(119909) =

1

21198881(119909 minus 119897

119903)2


1198872(1199062) =

1

21198882(1199062)2


(45)

1198873(119909) =

1

21198883(119909 minus 119897

119903)2

1198883isin (0infin) (46)


minus119905= minus

(119902119905)2

1198882

+ 1198883+ 1199021199052 (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+ 119902119905[minus119903119891minus119903119910

119905

119879

minus1

119905119910

119905+ (120590119890)2

+ (120590119894)2

] 1199021199051= 1198881

(47)

minus 119903

119905= minus

1198883(119909119903

119905minus 119897119903)

119902119905

minus 119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

minus [1199061

119905minus 119903119890

119905minus (120590119890)2

] minus (119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

)

minus (120590119894)2

119909119903(1199051) = 119897119903

(48)

minus 119897119905= minus119903119891119897119905+ 1198883(119909119903

119905minus 119897119903)2

minus 119902119905(120590119890)2

(119909119903

119905minus 1)2

minus 119902119905(120590119894)2

(119909119903

119905)2

119897 (1199051) = 0

(49)





1199062lowast

119905= minus

(119909 minus 119909119903

119905) 119902119905

1198882

1198922lowast

(119905 119909) = 1199062lowast

1198922lowast


lowast

119905= minus

(119909 minus 119909119903

119905)

119909minus1

119905119910

119905 1198923lowast

119879 timesX 997888rarr R119896

(50)

1198923119896lowast




(51)


V (119905 119909) = exp (minus119903119891(1199051minus 1199050)) [

1

2(119909 minus 119909

119903

119905)2

119902119905+

1

2119897119905] (52)

119869lowast= 119869 (119892

lowast) = E [V (119905

0 1199090)] (53)




minus119905= minus119887(119902

119905)2

+ 119888 + 2119886119905119902119905 1199021199051= 1198881

119887 =1

1198882 119888 = 119888

3isin (0infin) 119886 119879 997888rarr R

(54)


1199021

1199051= 119902 (119905

1minus 119905) 119886

1

119905= 119886 (119905

1minus 119905) 119902

1 [0 1199051minus 1199050] 997888rarr R

1

119905= minus (119905

1minus 119905) = minus119887(119902

1

119905)2

+ 119888 + 21198861

1199051199021

119905 1199021

1199050= 1198881

0 = 1

119905+ 119887(1199021

119905)2

minus 119888 minus 21198861

119905119902119905 1199021

0= 1198881

(55)


minus2

119905= minus119887(119902

2

119905)2

+ 119888 1199022

1199051= 1198881 119887 119888 isin (0infin) (56)


119905gt 0 for all 119905 isin 119879 = [0 119905

1minus 1199050]


1minus 1199050] 1199021119905

ge

1199022

119905gt 0 if


0 119887) ge (


119905

minus1198861

119905119887


(57)



(119886119905)2


1

1198882minus

1

1198882)

(58)


=


1minus 1199050]




1

1198882minus

1

1198882ge 0

(119886119905)2

le 1198884le (119888 minus 119888) (119887 minus 119887) = (119888

3minus 1198883) (

1

1198882minus

1

1198882)

(59)


1minus 1199050] 119902(1199051minus

119905) = 1199021

119905ge 1199022

119905gt 0




Note first that

V (119905 119909) = exp (minus119903119891(119905 minus 1199050)) [

1

2(119909 minus 119909

119903

119905)2

119902119905+

1

2119897119905]

119863119905V (119905 119909) = minus119903

119891V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))


119905) 119903

119905119902119905+

1

2(119909 minus 119909

119903

119905)2

119905+

1

2

119897119905]

119863119909V (119905 119909) = exp (minus119903

119891(119905 minus 1199050)) (119909 minus 119909

119903

119905) 119902119905

119863119909119909V (119905 119909) = exp (minus119903

119891(119905 minus 1199050)) 119902119905

(60)


So

0 = 119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))11986311990621198872(1199062)


119903

119905) 119902119905+ 11988821199062]

1199062lowast

= minus(119909 minus 119909

119903

119905) 119902119905

1198882

lowast= minus

(119909 minus 119909119903

119905) 119902119905

119909119902119905

minus1

119905119910

119905

(61)


1199021199051= 1198881 119909

119903(1199051) = 119897119903 119897 (119905

1) = 0

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050))

times [1

2(119909 minus 119909

119903(1199051))2

1199021199051+

1

2119897 (1199051)]

= exp (minus119903119891(1199051minus 1199050))

1

21198881(119909 minus 119897

119903)2

= exp (minus119903119891(1199051minus 1199050)) 1198871(119909)


exp (+119903119891(119905 minus 1199050))

times [119863119905V (119905 119909) +

1

2[(120590119890)2

(1 minus 119909)2

+(120590119894)2

(119909)2]119863119909119909V (119905 119909)

+ 119909 [119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]119863119909V (119905 119909)

+ [1199061

119905minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ 1199062lowast

119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050)) 1198872(1199062lowast

)

+ exp (minus119903119891(119905 minus 1199050)) 1198873(119909)

minus1

2

(119863119909V (119905 119909))

2

119863119909119909V (119905 119909)

119903119910

119905

119879

minus1

119905119910

119905]

= minus119903119891 1

2(119909 minus 119909

119903

119905)2

119902119905minus

1

2119903119891119897119905minus (119909 minus 119909

119903

119905) 119902119905119903

119905

+1

2(119909 minus 119909

119903

119905)2

119905+

1

2

119897119905

+1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2] 119902119905

+ (119909 minus 119909119903

119905) 119902119905[119909 (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+1199061

119905minus 119903119890

119905minus (120590119890)2

]

minus1

2

(119909 minus 119909119903

119905)2

(119902119905)2

1198882

+1

21198883(119909 minus 119897

119903)2

minus1

2(119909 minus 119909

119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

=1

2(119909)2[ minus 119903119891119902119905+ 119905+ (120590119890)2

119902119905+ (120590119894)2

119902119905

+ 2119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

minus(119902119905)2

1198882

+ 1198883minus 119902119905

119903119910

119905

119879

minus1

119905119910

119905]

+ 119909 [ + 119903119891119909119903

119905119902119905minus 119902119905119903

119905minus 119909119903

119905119905minus (120590119890)2

119902119905

minus 119909119903

119905119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ 119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] +119909119903

119905(119902119905)2

1198882

minus1198883119897119903+ 119909119903

119905119902119905

119903119910

119905

119879

minus1

119905119910

119905]

+1

2(119909)0[ minus 119903119891(119909119903

119905)2

119902119905+ 2119909119903

119905119902119905119903

119905+ (119909119903

119905)2

119905minus 119903119891119897119905

+ (120590119890)2

119902119905minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

]

minus(119909119903

119905)2

(119902119905)2

1198882

+ 1198883(119897119903)2

minus(119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905+ 119897119905]

(62)



minus 119902119905119903

119905minus 119909119903

119905119905+ 119903119891119909119903

119905119902119905minus 2(120590119890)2

119902119905

minus 119909119903

119905119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ 119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

]

+119909119903

119905(119902119905)2

1198882

minus 1198883119897119903+ 119909119903

119905119902119905

119903119910

119905

119879

minus1

119905119910

119905

= minus119902119905119903

119905

+ 119909119903

119905[minus

(119902119905)2

1198882

+ 1198883

+ 2119902119905(119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+119902119905((120590119890)2

+ (120590119894)2

minus 119903119891minus119903119910

119905

119879

minus1

119905119910

119905)]

minus 119909119903

119905119902119905[ (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

minus119903119891minus119903119910

119905

119879

minus1

119905119910

119905]

+ 119902 [minus(120590119890)2

+ (1199061

119905minus 119903119890

119905minus (120590119890)2

)] +119909119903

119905(119902119905)2

1198882

minus 1198883119897119903

= 119902119905[minus119903

119905+

1198883(119909119903

119905minus 119897119903)

119902119905

]

+ 119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ (119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

) + (120590119894)2

+ (1199061

119905minus 119903119890

119905minus (120590119890)2

)

= 0

(63)



119897119905minus 119903119891119897119905minus 119903119891(119909119903

119905)2

119902119905

+ 2119909119903

119905[119902119905119903

119905+ 119909119903

119905119905] minus (119909

119903

119905)2

119905+ (120590119890)2

119902119905

minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] + 1198883(119897119903)2

minus(119909119903

119905)2

(119902119905)2

1198882

minus (119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905


119897119905minus 119903119891119897119905minus 119903119891(119909119903

119905)2

119902119905+ (120590119890)2

119902119905

minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] + 1198883(119897119903)2

minus(119909119903

119905)2

(119902119905)2

1198882

minus (119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

+ 2119903119891(119909119903

119905)2

119902119905minus 2119909119903

119905119902119905(120590119890)2

minus 2(119909119903

119905)2

119902119905[119903

T+ 119903119890minus 119903119894+ (120590119890)2

+ (120590119894)2

]

+ 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] minus 11988831198971199032119909119903

119905

+2(119909119903

119905)2

(119902119905)2

1198882

+ 2(119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

minus(119909119903

119905)2

(119902119905)2

1198882

+ 1198883(119909119903

119905)2

+ (119909119903

119905)2

1199021199052 (119903

T+ 119903119890minus 119903119894+ (120590119890)2

+ (120590119894)2

)

+ (119909119903

119905)2

119902119905(minus119903119891+ (120590119890)2

+ (120590119894)2

minus119903119910

119905

119879

minus1

119905119910

119905)

= 119897119905minus 119903119891119897119905+ 1198883(119909119903

119905minus 119897119903)2

minus (120590119890)2

119902119905(119909119903

119905minus 1)2

minus (120590119894)2

119902119905(119909119903

119905)2

= 0

(64)







2gt 0 should


2(1199062) = (12)119888

2(1199062)2 given by (45) This







3(119909) = (12)119888



1198873(119909) = 119888

3[exp (119909 minus 119897

119903) + (119897119903minus 119909) minus 1] (65)







1199051198882 which





119903(119909) = (12)119888

3(119909 minus 119909




119903



119905119910



minus1

119905119910


deviation (119909 minus 119909119903

119905)




1199061

119905minus 119903119890

119905minus (120590119890)2

(66)



119890

119905is the


119903119890



1199051 119909119905lt 119897119903The term 119888

3(119909119903





119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

] (67)




T(119905) + (120590

119890)2+ (120590119894)2 is the effect of


(119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

) minus (120590119894)2

(68)




minus119903

119905= minus119896 (119909

119903

119905minus 119898) 119909

119903(1199051) = 119897119903

119896 = (119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2

+ (120590119894)2

)) +1198883

1199020

119898 =


) + (120590119890)2

(119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2+ (120590119894)2

)) + 11988831199020

(69)


119902119905

= 1199020 and lim

119905darr0119909119903




1199062lowast

119905= minus

(119909119905minus 119898) 119902

0

1198882

= gt 0 if 119909

119905lt 119898

lt 0 if 119909119905gt 119898

120587lowast

119905= minus

(119909119905minus 119898)

119909119905

119895

= gt 0 if 119909

119905lt 119898


(70)


Case 1 (119909119905











119898 =


) + (120590119890)2

(119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2+ (120590119894)2

)) + 11988831199020

(71)





Appendices













References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014



































119905119905ge0


0 1199051]





time 119905 with 1199092


119905and their









119905to denote

119909119905= [

1199091

119905

1199092

119905

] (3)


119897119905=

1199091

119905

1199092

119905

(4)




1

119905 In this case 1199061

119905119889119905 turns out to be


1199063

119905= 1199061

119905+ 1199062

119905 forall119905 (5)




119905119905ge0



119889119890119905= 119903119890

119905119889119905 + 120590

119890

119905119889119882119890

119905 119890 (119905

0) = 1198900 (6)




119890 Ω times 119879 rarr


119889ℎ119905= 119903ℎ

119905119889119905 + 120590

ℎ

119905119889119882ℎ

119905 ℎ (119905

0) = ℎ0 (7)







119896

119905 119896 isin N

119899=


119910 = (T (119905) 1199101

119905 119910

119899

119905) (8)


119905 119910

119899


model 119910 as

119889119910119905= 119903119910

119905119889119905 + Σ

119910

119905119889119882119910

119905 119910 (119905

0) = 1199100 (9)





119878 = 120587 isin R119899+1

120587 = (1205870 120587

119899)119879

1205870+ sdot sdot sdot + 120587

119899= 1

1205870ge 0 120587

119899ge 0

(10)


119889ℎ119905= 120587119879

119905119889119910119905= 120587119879

119905119903119910

119905119889119905 + 120587

119879

119905Σ119910

119905119889119882119910

119905 (11)


119903T(119905) = 119903

1199100

(119905) 119903T 119879 997888rarr R

+


119903119910

119905= (119903

T(119905)

119903119910

119905

119879

+ 119903T(119905) 1119899)

119879

119910 119879 997888rarr R

119899

120587119905= (1205870

119905 119879

119905)119879

= (1205870

119905 1205871

119905 120587

119896

119905)119879

119879 997888rarr R119896

Σ119910

119905= (

0 sdot sdot sdot 0

Σ119910

119905

) Σ119910

119905isin R119899times119899

119905= Σ119910

119905

Σ119910

119905

119879

Thenwe have that

120587119879

119905119903119910

119905= 1205870

119905119903T(119905) +

119895119879

119905119910

119905+ 119895119879

119905119903T(119905) 1119899

= 119903T(119905) +

119879

119905119910

119905

120587119879

119905Σ119910

119905119889119882119910

119905= 119879

119905Σ119910

119905119889119882119910

119905

119889ℎ119905= [119903

T(119905) +

119879

119905119910

119905] 119889119905 +

119879

119905Σ119910

119905119889119882119910

119905 ℎ (119905

0) = ℎ0

(12)






119889119894119905= 119903119894

119905119889119905 + 120590

119894119889119882119894

119905 119894 (119905

0) = 1198940 (13)




1198891199091

119905= 1199091

119905119889ℎ119905+ 1199092

1199051199063

119905119889119905 minus 119909

2

119905119889119890119905

= [119903T(119905) 1199091

119905+ 1199091

119905119879

119905119910

119905+ 1199092

1199051199061

119905+ 1199092

1199051199062

119905minus 1199092

119905119903119890

119905] 119889119905

+ [1199091

119905119879

119905Σ119910

119905119889119882119910

119905minus 1199092

119905120590119890119889119882119890

119905]

1198891199092

119905= 1199092

119905119889119894119905minus 1199092

119905119889119890119905

= 1199092

119905[119903119894

119905119889119905 + 120590

119894119889119882119894

119905] minus 1199092

119905[119903119890

119905119889119905 + 120590

119890119889119882119890

119905]

= 1199092

119905[119903119894

119905minus 119903119890

119905] 119889119905 + 119909

2

119905[120590119894119889119882119894

119905minus 120590119890119889119882119890

119905]

(14)



119889119909119905= 119860119905119909119905119889119905 + 119873 (119909

119905) 119906119905119889119905 + 119886

119905119889119905 + 119878 (119909

119905 119906119905) 119889119882119905 (15)


119906119905= [

1199062

119905

119905

] 119906 Ω times 119879 997888rarr R119899+1

119860119905= [

119903T(119905) minus119903

119890

119905

0 119903119894

119905minus 119903119890

119905

]

119873 (119909119905) = [

1199092

1199051199091

119905

119903119910

119905

119879

0 0

] 119886119905= [

1199092

1199051199061

119905

0]

119878 (119909119905 119906119905) = [

1199091

119905119879

119905Σ119910

119905minus1199092

119905120590119890

0

0 minus1199092

119905120590119890

1199092

119905120590119894]

119882119905= [

[

119882119910

119905

119882119890

119905

119882119894

119905

]

]

(16)


where 119882119910

119905 119882119890119905 and 119882

119894



119905gt 0 120590119894

119905gt 0 and




119873(119909119905) 119906119905= [

1199092

119905

0] 1199062

119905+ [

1199091

119905

119903119910

119905

119879

0

] 119905

= [0 1

0 0] 1199091199051199063

119905+

119899

sum

119898=1

[1199091

119905119910119898

119905

0] 119898

119905

= 11986101199091199051199060

119905+

119899

sum

119898=1

[119910119898

1199050

0 0] 119909119905119898

119905

=

119899

sum

119898=0

[119861119898119909119905] 119906119898

119905

119878 (119909119905 119906119905) 119889119882119905= [

[119879

119905119905119905]12

0

0 0

] 1199091199051198891198821

119905

+ [0 minus120590119890

0 minus120590119890] 1199091199051198891198822

119905+ [

0 0

0 120590119894] 1199091199051198891198823

119905

=

3

sum

119895=1

[119872119895119895(119906119905) 119909119905] 119889119882119895119895

119905

(17)






of 1199091119905via the term 119909

1

119905

119903119910

119905

119879



119889119909119905= 119860119905119909119905119889119905 +

119899

sum

119895=0

[119861119895119909119905] 119906119895

119905119889119905 + 119886

119905119889119905

+

3

sum

119895=1

[119872119895(119906119905) 119909119905] 119889119882119895119895

119905

(18)


119905=

1199091

1199051199092

119905


119889119909119905= 119909119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

+119903119910

119905

119879

119905] 119889119905

+ [1199061

119905+ 1199062

119905minus 119903119890

119905minus (120590119890)2

] 119889119905

+ [(120590119890)2

(1 minus 119909119905)2

+ (120590119894)2

1199092

119905+ 1199092

119905119879

119905119905119905]

12

119889119882119905

119909 (1199050) = 1199090

(19)


1 1199092)119879 as in (14) Next we

determine

119889(1199092

119905)minus1

= minus(1199092

119905)minus2

1198891199092

119905+

1

22(1199092)minus3

119905119889 lt 119909

2 1199092gt 119905


119905(119903119894

119905minus 119903119890

119905) + (119909

2

119905)minus1

((120590119890)2

+ (120590119894)2

)] 119889119905

minus (1199092

119905)minus1

[0 minus120590119890

120590119894] 119889119882119905

119889119909119905= 1199091

119905119889(1199092

119905)minus1

+ (1199092

119905)minus1

1198891199091

119905+ 119889 lt 119909

1 (119909

2)minus1

gt 119905

= [119903T(119905) 119909119905minus 119903119890

119905+ 1199061

119905+ 1199062

119905+ 119909119905119910

119905119905

minus119909119905[119903119894

119905minus 119903119890

119905] + 119909119905((120590119890)2

+ (120590119894)2

) minus (120590119890)2

] 119889119905

+ (119909119905119879

119905Σ119910

119905minus 120590119890(1 minus 119909

119905) minus 120590119894119909119905) 119889119882119905

= 119909119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

+119903119910

119905

119879

119905] 119889119905

+ [1199061

119905+ 1199062

119905minus 119903119890

119905minus (120590119890)2

] 119889119905

+ [(120590119890)2

(1 minus 119909119905)2

+ (120590119894)2

(119909119905)2

+(119909119905)2

119879

119905119905119905+ (120590119894)2

(119909119905)2

]

12

119889119882119905

(20)


minus119903119890

119905+ 119909119905119903119890

119905= minus119903119890

119905(119909119905minus 1) (21)


119890)2+ 119909119905(120590119890)2= (120590119890)2(119909119905minus 1) appears




Time (years)


0 01 02 03 04 05 06 07 08 09 1

5

0

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

minus45



Parameter Value119862 500119903119894 001119910 0061199062 003

119903119862 005120590119890 16

04 150119903119890 004120590119894 18

1199061 001

119882 003











119905in (4)















(22)


Table4Summaryof

netstablefun

ding

ratio

Availables

tablefun

ding

(sou

rces)

Requ

iredstablefund

ing(uses)

Item

Avlblty

Fctr

Item

Rqrd

Fctr

(i)T1KandT2

Kinstr

uments

(ii)O

ther

preferredshares

andcapitalinstrum

entsin

excessof

T2Kallowableam

ount

having

aneffectiv

ematurity

of1Y

rorgt

1Yr

(iii)Other

liabilitiesw

ithan

effectiv

ematurity

of1Y

rorgt

1Yr

100

(i)Ca

sh(ii)S

hort-te

rmun

securedactiv

elytraded

instr

uments(lt1Y

r)(iii)Securitiesw

ithexactly

offsetting

reverser

epo

(iv)S

ecurities

with

remaining

maturity

lt1Y

r(v)N


ancialsw

ithremaining

maturity

lt1Y

r

0

Stabledepo

sitso

fretailand

smallbusinessc

ustomers

(non

maturity

orresid

ualm

aturity

lt1Y

r)90

Debtissuedor

guaranteed


sBISIM

FEC

non

centralgovernm


entb

anks

with

a0ris

kweightu

nder

BaselIIS

tand

ardizedAp

proach

5

Lessstabledepo

sitso

fretailand

smallbusinessc

ustomers

(non

maturity

orresid

ualm

aturity

lt1Y

r)80

Unencum

beredno

nfinancialsenioru

nsecured

corporateb

onds

and

coveredbo

ndsrated

atleastA

Aminusanddebt

thatisissuedby

SovereignsC

entralBa

nksandPS

Eswith

arisk

weightin

gof

20

maturity

ge1Y

r

20

Who

lesalefund

ingprovided

byno

nfinancialcorpo

ratecusto

mers


ultilateraldevelopm

entb

anksand

PSEs

(non

maturity

orresid

ualm

aturity

lt1Y

r)50

(i)Unencum


rnon

financialsenior

unsecuredcorporateb

onds

(orc

overed

bond

s)ratedfro

mA+to

Aminus

maturity

ge1Y

r(ii)G

old

(iii)Lo

anstono

nfinancialcorpo

rateclients

SovereignsC

entral

Bank

sandPS

Eswith

amaturity

lt1yr

50


ndequityno

tincludedabove

0

Unencum

beredresid

entia

lmortgages

ofanymaturity

andother


ding

loanstofin

ancialinstitu

tions

with

aremaining

maturity

of1Y

rorgt

1Yrthatw

ould

qualify

forthe

35or

lower

riskweightu

nder

baselIIstand

ardizedapproach

forc

redit

risk

65

Other


allbusinessesh

avingam

aturity

lt1Y

r85

Allothera

ssets

100

Offbalances

heetexpo

sures

Und

rawnam

ount

ofcommitted

creditandliq

uidityfacilities

5

Other

contingent

fund

ingob

ligations

National

superviso

rydiscretio

n






5



50











119889119909119905= 119860119905119909119905119889119905 +

119899

sum

119895=0

119861119895119909119905119892119895(119905 119909119905) 119889119905 + 119886

119905119889119905

+

3

sum

119895=1

119872119895119895(119892 (119905 119909

119905)) 119909119905119889119882119895119895

119905 119909 (119905

0) = 1199090

(23)



119869 (119892) = E [int

1199051

1199050

exp (minus119903119891(119897 minus 1199050)) 119887 (119897 119909

119897 119892 (119897 119909

119897)) 119889119897

+ exp (minus119903119891(1199051minus 1199050)) 1198871(119909 (1199051)) ]

(24)


where 119892 isin G119860 119879 = [119905

0 1199051] and 119887



119887 (119905 119909 119906) = 1198872(1199062) + 1198873(1199091

1199092) (25)

for 1198872 U2rarr R+ and 119887





= 0







119869 (119892) (26)



119869 (119892) (27)



119892isinG119860

119869 (119892) isin G119860 (28)





119869 (119892)


119869 (119892)

(29)

119869 (119892) = E [int

1199051

1199050

exp (minus119903119891(119897 minus 1199050)) [1198872(1199062

119905) + 1198873(119909119905)] d119905

+ exp (minus119903119891(1199051minus 1199050)) 1198871(119909 (1199051)) ]

(30)


3 R+ rarr R+





1198872(1199062) isin 1198622(R)


11986311990621198872(1199062) = minusinfin


11986311990621198872(1199062) = +infin

119863119906211990621198872(1199062) gt 0 forall119906

2isin R

(31)


2



0 = 119863119905V (119905 119909) +

1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909119905)2

]119863119909119909V (119905 119909)

+ 119909 (119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)119863119909V (119905 119909)

+ [1199061

119905minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ 1199062

119905

lowast

119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050)) 1198872(1199062

119905

lowast

)


[119863119909V (119905 119909)]

2

2119863119909119909V (119905 119909)

119903119910

119905

119879

minus1

119905119910

119905

(32)

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050)) 1198871(119909) (33)


0 = 119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))11986311990621198872(1199062

119905) (34)


1198922lowast

(119905 119909) = 1199062lowast

1198922lowast


with 1199062lowast


lowast= minus

119863119909V (119905 119909)

119909119863119909119909V (119905 119909)

minus1

119905119910

119905

1198923119896lowast



(36)



119869lowast= 119869 (119892

lowast) = E [V (119905 119909

0)] (37)



12(119879timesX)

is given by

0 = 119863119905119908 (119905 119909)


[1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2

+(119909)2119879119905]119863119909119909119908 (119905 119909)

+ [ (119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

+119903119910

119905

119879

) 119909]119863119909119908 (119905 119909)

+ [1199061

119905+ 1199062minus 119903119890

119905minus (120590119890)2

]119863119909119908 (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) [1198872(1199062) + 1198873(119909)] ]

119908 (1199051 119909) = exp (minus119903

119891(1199051minus 1199050)) 1198871(119909)

(38)



(b) depending on




inf1199062isinR

1198672(119905 119909 119906

2)

1198672(119905 119909 119906) = 119906

2119863119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) 1198872(1199062)

11986311990621198672(119905 119909 119906) = 119863

119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050))11986311990621198872(1199062) = 0

(39)


2lowast

isin U = R

and

inf1199062isinR

1198672(119905 119909 119906

2) = 119867

2(119905 119909 119906

2lowast

)

= 1199062lowast

119863119909V (119905 119909)


)

(40)




infisinR119896

1198673(119905 119909 )

1198673(119905 119909 ) =

1

2[119879

119905119905119905] (119909)2119863119909119909V (119905 119909)

+119903119910

119905

119879

119909119863119909V (119905 119909)

=1

2(119905+ (

119909 [119863119909V (119905 119909)]

(119909)2119863119909119909V (119905 119909)

) minus1

119905119910

119905)

119879

times 119905(119909)2119863119909119909V (119905 119909)

times (119905+ (

119909 [119863119909V (119905 119909)]

(119909)2119863119909119909V (119905 119909)

) minus1

119905119910

119905)

minus1

2(

[119909119863119909V (119905 119909)]

2

(119909)2119863119909119909V (119905 119909)

)119903119910

119905

119879

minus1

119905119910

119905

119905(119909)2119863119909119909V (119905 119909) gt 0 forall119909 isin R 0

(41)

Hence we have that

lowast= minus

119863119909V (119905 119909)

119909119863119909119909V (119905 119909)

minus1

119905119910

119905

1198923119896lowast

(119905 119909) =

119896lowast if

119896

sum

119894=1

119894lowast

isin [0 1]

119896lowast

sum119896

119895=1119895lowast

if119896

sum

119894=1

119894lowast

gt 1


1198673(119905 119909

lowast) = minus

[119863119909V (119905 119909)]

2

2119863119909119909V (119905 119909)

(119910

119905)119879

minus1

119905119910

119905

(42)





0 = 119863119905V (119905 119909)


1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2

+(119909)2(119879119905) ]119863

119909119909V (119905 119909)

+ [119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+(120590119894)2

+119903119910

119905

119879

] 119909

+ [1199061

119905+ 1199062minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) [1198872(1199062) + 1198873(119909)]

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050)) 1198871(119909)

(43)



lowast=

119869(119892lowast) = E[V(119905

0 1199090)]



119869 (119892) = E [int

1199051

1199050

exp (minus119903119891(119897 minus 1199050))

times [1

21198882((1199062)2

(119897)) +1

21198883(119909119905minus 119897119903)2

] 119889119897

+1

21198881(119909 (1199051) minus 119897119903)2 exp (minus119903

119891(1199051minus 1199050)) ]

(44)


1198871(119909) =

1

21198881(119909 minus 119897

119903)2


1198872(1199062) =

1

21198882(1199062)2


(45)

1198873(119909) =

1

21198883(119909 minus 119897

119903)2

1198883isin (0infin) (46)


minus119905= minus

(119902119905)2

1198882

+ 1198883+ 1199021199052 (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+ 119902119905[minus119903119891minus119903119910

119905

119879

minus1

119905119910

119905+ (120590119890)2

+ (120590119894)2

] 1199021199051= 1198881

(47)

minus 119903

119905= minus

1198883(119909119903

119905minus 119897119903)

119902119905

minus 119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

minus [1199061

119905minus 119903119890

119905minus (120590119890)2

] minus (119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

)

minus (120590119894)2

119909119903(1199051) = 119897119903

(48)

minus 119897119905= minus119903119891119897119905+ 1198883(119909119903

119905minus 119897119903)2

minus 119902119905(120590119890)2

(119909119903

119905minus 1)2

minus 119902119905(120590119894)2

(119909119903

119905)2

119897 (1199051) = 0

(49)





1199062lowast

119905= minus

(119909 minus 119909119903

119905) 119902119905

1198882

1198922lowast

(119905 119909) = 1199062lowast

1198922lowast


lowast

119905= minus

(119909 minus 119909119903

119905)

119909minus1

119905119910

119905 1198923lowast

119879 timesX 997888rarr R119896

(50)

1198923119896lowast




(51)


V (119905 119909) = exp (minus119903119891(1199051minus 1199050)) [

1

2(119909 minus 119909

119903

119905)2

119902119905+

1

2119897119905] (52)

119869lowast= 119869 (119892

lowast) = E [V (119905

0 1199090)] (53)




minus119905= minus119887(119902

119905)2

+ 119888 + 2119886119905119902119905 1199021199051= 1198881

119887 =1

1198882 119888 = 119888

3isin (0infin) 119886 119879 997888rarr R

(54)


1199021

1199051= 119902 (119905

1minus 119905) 119886

1

119905= 119886 (119905

1minus 119905) 119902

1 [0 1199051minus 1199050] 997888rarr R

1

119905= minus (119905

1minus 119905) = minus119887(119902

1

119905)2

+ 119888 + 21198861

1199051199021

119905 1199021

1199050= 1198881

0 = 1

119905+ 119887(1199021

119905)2

minus 119888 minus 21198861

119905119902119905 1199021

0= 1198881

(55)


minus2

119905= minus119887(119902

2

119905)2

+ 119888 1199022

1199051= 1198881 119887 119888 isin (0infin) (56)


119905gt 0 for all 119905 isin 119879 = [0 119905

1minus 1199050]


1minus 1199050] 1199021119905

ge

1199022

119905gt 0 if


0 119887) ge (


119905

minus1198861

119905119887


(57)



(119886119905)2


1

1198882minus

1

1198882)

(58)


=


1minus 1199050]




1

1198882minus

1

1198882ge 0

(119886119905)2

le 1198884le (119888 minus 119888) (119887 minus 119887) = (119888

3minus 1198883) (

1

1198882minus

1

1198882)

(59)


1minus 1199050] 119902(1199051minus

119905) = 1199021

119905ge 1199022

119905gt 0




Note first that

V (119905 119909) = exp (minus119903119891(119905 minus 1199050)) [

1

2(119909 minus 119909

119903

119905)2

119902119905+

1

2119897119905]

119863119905V (119905 119909) = minus119903

119891V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))


119905) 119903

119905119902119905+

1

2(119909 minus 119909

119903

119905)2

119905+

1

2

119897119905]

119863119909V (119905 119909) = exp (minus119903

119891(119905 minus 1199050)) (119909 minus 119909

119903

119905) 119902119905

119863119909119909V (119905 119909) = exp (minus119903

119891(119905 minus 1199050)) 119902119905

(60)


So

0 = 119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))11986311990621198872(1199062)


119903

119905) 119902119905+ 11988821199062]

1199062lowast

= minus(119909 minus 119909

119903

119905) 119902119905

1198882

lowast= minus

(119909 minus 119909119903

119905) 119902119905

119909119902119905

minus1

119905119910

119905

(61)


1199021199051= 1198881 119909

119903(1199051) = 119897119903 119897 (119905

1) = 0

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050))

times [1

2(119909 minus 119909

119903(1199051))2

1199021199051+

1

2119897 (1199051)]

= exp (minus119903119891(1199051minus 1199050))

1

21198881(119909 minus 119897

119903)2

= exp (minus119903119891(1199051minus 1199050)) 1198871(119909)


exp (+119903119891(119905 minus 1199050))

times [119863119905V (119905 119909) +

1

2[(120590119890)2

(1 minus 119909)2

+(120590119894)2

(119909)2]119863119909119909V (119905 119909)

+ 119909 [119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]119863119909V (119905 119909)

+ [1199061

119905minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ 1199062lowast

119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050)) 1198872(1199062lowast

)

+ exp (minus119903119891(119905 minus 1199050)) 1198873(119909)

minus1

2

(119863119909V (119905 119909))

2

119863119909119909V (119905 119909)

119903119910

119905

119879

minus1

119905119910

119905]

= minus119903119891 1

2(119909 minus 119909

119903

119905)2

119902119905minus

1

2119903119891119897119905minus (119909 minus 119909

119903

119905) 119902119905119903

119905

+1

2(119909 minus 119909

119903

119905)2

119905+

1

2

119897119905

+1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2] 119902119905

+ (119909 minus 119909119903

119905) 119902119905[119909 (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+1199061

119905minus 119903119890

119905minus (120590119890)2

]

minus1

2

(119909 minus 119909119903

119905)2

(119902119905)2

1198882

+1

21198883(119909 minus 119897

119903)2

minus1

2(119909 minus 119909

119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

=1

2(119909)2[ minus 119903119891119902119905+ 119905+ (120590119890)2

119902119905+ (120590119894)2

119902119905

+ 2119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

minus(119902119905)2

1198882

+ 1198883minus 119902119905

119903119910

119905

119879

minus1

119905119910

119905]

+ 119909 [ + 119903119891119909119903

119905119902119905minus 119902119905119903

119905minus 119909119903

119905119905minus (120590119890)2

119902119905

minus 119909119903

119905119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ 119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] +119909119903

119905(119902119905)2

1198882

minus1198883119897119903+ 119909119903

119905119902119905

119903119910

119905

119879

minus1

119905119910

119905]

+1

2(119909)0[ minus 119903119891(119909119903

119905)2

119902119905+ 2119909119903

119905119902119905119903

119905+ (119909119903

119905)2

119905minus 119903119891119897119905

+ (120590119890)2

119902119905minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

]

minus(119909119903

119905)2

(119902119905)2

1198882

+ 1198883(119897119903)2

minus(119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905+ 119897119905]

(62)



minus 119902119905119903

119905minus 119909119903

119905119905+ 119903119891119909119903

119905119902119905minus 2(120590119890)2

119902119905

minus 119909119903

119905119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ 119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

]

+119909119903

119905(119902119905)2

1198882

minus 1198883119897119903+ 119909119903

119905119902119905

119903119910

119905

119879

minus1

119905119910

119905

= minus119902119905119903

119905

+ 119909119903

119905[minus

(119902119905)2

1198882

+ 1198883

+ 2119902119905(119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+119902119905((120590119890)2

+ (120590119894)2

minus 119903119891minus119903119910

119905

119879

minus1

119905119910

119905)]

minus 119909119903

119905119902119905[ (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

minus119903119891minus119903119910

119905

119879

minus1

119905119910

119905]

+ 119902 [minus(120590119890)2

+ (1199061

119905minus 119903119890

119905minus (120590119890)2

)] +119909119903

119905(119902119905)2

1198882

minus 1198883119897119903

= 119902119905[minus119903

119905+

1198883(119909119903

119905minus 119897119903)

119902119905

]

+ 119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ (119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

) + (120590119894)2

+ (1199061

119905minus 119903119890

119905minus (120590119890)2

)

= 0

(63)



119897119905minus 119903119891119897119905minus 119903119891(119909119903

119905)2

119902119905

+ 2119909119903

119905[119902119905119903

119905+ 119909119903

119905119905] minus (119909

119903

119905)2

119905+ (120590119890)2

119902119905

minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] + 1198883(119897119903)2

minus(119909119903

119905)2

(119902119905)2

1198882

minus (119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905


119897119905minus 119903119891119897119905minus 119903119891(119909119903

119905)2

119902119905+ (120590119890)2

119902119905

minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] + 1198883(119897119903)2

minus(119909119903

119905)2

(119902119905)2

1198882

minus (119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

+ 2119903119891(119909119903

119905)2

119902119905minus 2119909119903

119905119902119905(120590119890)2

minus 2(119909119903

119905)2

119902119905[119903

T+ 119903119890minus 119903119894+ (120590119890)2

+ (120590119894)2

]

+ 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] minus 11988831198971199032119909119903

119905

+2(119909119903

119905)2

(119902119905)2

1198882

+ 2(119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

minus(119909119903

119905)2

(119902119905)2

1198882

+ 1198883(119909119903

119905)2

+ (119909119903

119905)2

1199021199052 (119903

T+ 119903119890minus 119903119894+ (120590119890)2

+ (120590119894)2

)

+ (119909119903

119905)2

119902119905(minus119903119891+ (120590119890)2

+ (120590119894)2

minus119903119910

119905

119879

minus1

119905119910

119905)

= 119897119905minus 119903119891119897119905+ 1198883(119909119903

119905minus 119897119903)2

minus (120590119890)2

119902119905(119909119903

119905minus 1)2

minus (120590119894)2

119902119905(119909119903

119905)2

= 0

(64)







2gt 0 should


2(1199062) = (12)119888

2(1199062)2 given by (45) This







3(119909) = (12)119888



1198873(119909) = 119888

3[exp (119909 minus 119897

119903) + (119897119903minus 119909) minus 1] (65)







1199051198882 which





119903(119909) = (12)119888

3(119909 minus 119909




119903



119905119910



minus1

119905119910


deviation (119909 minus 119909119903

119905)




1199061

119905minus 119903119890

119905minus (120590119890)2

(66)



119890

119905is the


119903119890



1199051 119909119905lt 119897119903The term 119888

3(119909119903





119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

] (67)




T(119905) + (120590

119890)2+ (120590119894)2 is the effect of


(119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

) minus (120590119894)2

(68)




minus119903

119905= minus119896 (119909

119903

119905minus 119898) 119909

119903(1199051) = 119897119903

119896 = (119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2

+ (120590119894)2

)) +1198883

1199020

119898 =


) + (120590119890)2

(119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2+ (120590119894)2

)) + 11988831199020

(69)


119902119905

= 1199020 and lim

119905darr0119909119903




1199062lowast

119905= minus

(119909119905minus 119898) 119902

0

1198882

= gt 0 if 119909

119905lt 119898

lt 0 if 119909119905gt 119898

120587lowast

119905= minus

(119909119905minus 119898)

119909119905

119895

= gt 0 if 119909

119905lt 119898


(70)


Case 1 (119909119905











119898 =


) + (120590119890)2

(119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2+ (120590119894)2

)) + 11988831199020

(71)





Appendices













References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014










119905to denote

119909119905= [

1199091

119905

1199092

119905

] (3)


119897119905=

1199091

119905

1199092

119905

(4)




1

119905 In this case 1199061

119905119889119905 turns out to be


1199063

119905= 1199061

119905+ 1199062

119905 forall119905 (5)




119905119905ge0



119889119890119905= 119903119890

119905119889119905 + 120590

119890

119905119889119882119890

119905 119890 (119905

0) = 1198900 (6)




119890 Ω times 119879 rarr


119889ℎ119905= 119903ℎ

119905119889119905 + 120590

ℎ

119905119889119882ℎ

119905 ℎ (119905

0) = ℎ0 (7)







119896

119905 119896 isin N

119899=


119910 = (T (119905) 1199101

119905 119910

119899

119905) (8)


119905 119910

119899


model 119910 as

119889119910119905= 119903119910

119905119889119905 + Σ

119910

119905119889119882119910

119905 119910 (119905

0) = 1199100 (9)





119878 = 120587 isin R119899+1

120587 = (1205870 120587

119899)119879

1205870+ sdot sdot sdot + 120587

119899= 1

1205870ge 0 120587

119899ge 0

(10)


119889ℎ119905= 120587119879

119905119889119910119905= 120587119879

119905119903119910

119905119889119905 + 120587

119879

119905Σ119910

119905119889119882119910

119905 (11)


119903T(119905) = 119903

1199100

(119905) 119903T 119879 997888rarr R

+


119903119910

119905= (119903

T(119905)

119903119910

119905

119879

+ 119903T(119905) 1119899)

119879

119910 119879 997888rarr R

119899

120587119905= (1205870

119905 119879

119905)119879

= (1205870

119905 1205871

119905 120587

119896

119905)119879

119879 997888rarr R119896

Σ119910

119905= (

0 sdot sdot sdot 0

Σ119910

119905

) Σ119910

119905isin R119899times119899

119905= Σ119910

119905

Σ119910

119905

119879

Thenwe have that

120587119879

119905119903119910

119905= 1205870

119905119903T(119905) +

119895119879

119905119910

119905+ 119895119879

119905119903T(119905) 1119899

= 119903T(119905) +

119879

119905119910

119905

120587119879

119905Σ119910

119905119889119882119910

119905= 119879

119905Σ119910

119905119889119882119910

119905

119889ℎ119905= [119903

T(119905) +

119879

119905119910

119905] 119889119905 +

119879

119905Σ119910

119905119889119882119910

119905 ℎ (119905

0) = ℎ0

(12)






119889119894119905= 119903119894

119905119889119905 + 120590

119894119889119882119894

119905 119894 (119905

0) = 1198940 (13)




1198891199091

119905= 1199091

119905119889ℎ119905+ 1199092

1199051199063

119905119889119905 minus 119909

2

119905119889119890119905

= [119903T(119905) 1199091

119905+ 1199091

119905119879

119905119910

119905+ 1199092

1199051199061

119905+ 1199092

1199051199062

119905minus 1199092

119905119903119890

119905] 119889119905

+ [1199091

119905119879

119905Σ119910

119905119889119882119910

119905minus 1199092

119905120590119890119889119882119890

119905]

1198891199092

119905= 1199092

119905119889119894119905minus 1199092

119905119889119890119905

= 1199092

119905[119903119894

119905119889119905 + 120590

119894119889119882119894

119905] minus 1199092

119905[119903119890

119905119889119905 + 120590

119890119889119882119890

119905]

= 1199092

119905[119903119894

119905minus 119903119890

119905] 119889119905 + 119909

2

119905[120590119894119889119882119894

119905minus 120590119890119889119882119890

119905]

(14)



119889119909119905= 119860119905119909119905119889119905 + 119873 (119909

119905) 119906119905119889119905 + 119886

119905119889119905 + 119878 (119909

119905 119906119905) 119889119882119905 (15)


119906119905= [

1199062

119905

119905

] 119906 Ω times 119879 997888rarr R119899+1

119860119905= [

119903T(119905) minus119903

119890

119905

0 119903119894

119905minus 119903119890

119905

]

119873 (119909119905) = [

1199092

1199051199091

119905

119903119910

119905

119879

0 0

] 119886119905= [

1199092

1199051199061

119905

0]

119878 (119909119905 119906119905) = [

1199091

119905119879

119905Σ119910

119905minus1199092

119905120590119890

0

0 minus1199092

119905120590119890

1199092

119905120590119894]

119882119905= [

[

119882119910

119905

119882119890

119905

119882119894

119905

]

]

(16)


where 119882119910

119905 119882119890119905 and 119882

119894



119905gt 0 120590119894

119905gt 0 and




119873(119909119905) 119906119905= [

1199092

119905

0] 1199062

119905+ [

1199091

119905

119903119910

119905

119879

0

] 119905

= [0 1

0 0] 1199091199051199063

119905+

119899

sum

119898=1

[1199091

119905119910119898

119905

0] 119898

119905

= 11986101199091199051199060

119905+

119899

sum

119898=1

[119910119898

1199050

0 0] 119909119905119898

119905

=

119899

sum

119898=0

[119861119898119909119905] 119906119898

119905

119878 (119909119905 119906119905) 119889119882119905= [

[119879

119905119905119905]12

0

0 0

] 1199091199051198891198821

119905

+ [0 minus120590119890

0 minus120590119890] 1199091199051198891198822

119905+ [

0 0

0 120590119894] 1199091199051198891198823

119905

=

3

sum

119895=1

[119872119895119895(119906119905) 119909119905] 119889119882119895119895

119905

(17)






of 1199091119905via the term 119909

1

119905

119903119910

119905

119879



119889119909119905= 119860119905119909119905119889119905 +

119899

sum

119895=0

[119861119895119909119905] 119906119895

119905119889119905 + 119886

119905119889119905

+

3

sum

119895=1

[119872119895(119906119905) 119909119905] 119889119882119895119895

119905

(18)


119905=

1199091

1199051199092

119905


119889119909119905= 119909119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

+119903119910

119905

119879

119905] 119889119905

+ [1199061

119905+ 1199062

119905minus 119903119890

119905minus (120590119890)2

] 119889119905

+ [(120590119890)2

(1 minus 119909119905)2

+ (120590119894)2

1199092

119905+ 1199092

119905119879

119905119905119905]

12

119889119882119905

119909 (1199050) = 1199090

(19)


1 1199092)119879 as in (14) Next we

determine

119889(1199092

119905)minus1

= minus(1199092

119905)minus2

1198891199092

119905+

1

22(1199092)minus3

119905119889 lt 119909

2 1199092gt 119905


119905(119903119894

119905minus 119903119890

119905) + (119909

2

119905)minus1

((120590119890)2

+ (120590119894)2

)] 119889119905

minus (1199092

119905)minus1

[0 minus120590119890

120590119894] 119889119882119905

119889119909119905= 1199091

119905119889(1199092

119905)minus1

+ (1199092

119905)minus1

1198891199091

119905+ 119889 lt 119909

1 (119909

2)minus1

gt 119905

= [119903T(119905) 119909119905minus 119903119890

119905+ 1199061

119905+ 1199062

119905+ 119909119905119910

119905119905

minus119909119905[119903119894

119905minus 119903119890

119905] + 119909119905((120590119890)2

+ (120590119894)2

) minus (120590119890)2

] 119889119905

+ (119909119905119879

119905Σ119910

119905minus 120590119890(1 minus 119909

119905) minus 120590119894119909119905) 119889119882119905

= 119909119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

+119903119910

119905

119879

119905] 119889119905

+ [1199061

119905+ 1199062

119905minus 119903119890

119905minus (120590119890)2

] 119889119905

+ [(120590119890)2

(1 minus 119909119905)2

+ (120590119894)2

(119909119905)2

+(119909119905)2

119879

119905119905119905+ (120590119894)2

(119909119905)2

]

12

119889119882119905

(20)


minus119903119890

119905+ 119909119905119903119890

119905= minus119903119890

119905(119909119905minus 1) (21)


119890)2+ 119909119905(120590119890)2= (120590119890)2(119909119905minus 1) appears




Time (years)


0 01 02 03 04 05 06 07 08 09 1

5

0

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

minus45



Parameter Value119862 500119903119894 001119910 0061199062 003

119903119862 005120590119890 16

04 150119903119890 004120590119894 18

1199061 001

119882 003











119905in (4)















(22)


Table4Summaryof

netstablefun

ding

ratio

Availables

tablefun

ding

(sou

rces)

Requ

iredstablefund

ing(uses)

Item

Avlblty

Fctr

Item

Rqrd

Fctr

(i)T1KandT2

Kinstr

uments

(ii)O

ther

preferredshares

andcapitalinstrum

entsin

excessof

T2Kallowableam

ount

having

aneffectiv

ematurity

of1Y

rorgt

1Yr

(iii)Other

liabilitiesw

ithan

effectiv

ematurity

of1Y

rorgt

1Yr

100

(i)Ca

sh(ii)S

hort-te

rmun

securedactiv

elytraded

instr

uments(lt1Y

r)(iii)Securitiesw

ithexactly

offsetting

reverser

epo

(iv)S

ecurities

with

remaining

maturity

lt1Y

r(v)N


ancialsw

ithremaining

maturity

lt1Y

r

0

Stabledepo

sitso

fretailand

smallbusinessc

ustomers

(non

maturity

orresid

ualm

aturity

lt1Y

r)90

Debtissuedor

guaranteed


sBISIM

FEC

non

centralgovernm


entb

anks

with

a0ris

kweightu

nder

BaselIIS

tand

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5



50











119889119909119905= 119860119905119909119905119889119905 +

119899

sum

119895=0

119861119895119909119905119892119895(119905 119909119905) 119889119905 + 119886

119905119889119905

+

3

sum

119895=1

119872119895119895(119892 (119905 119909

119905)) 119909119905119889119882119895119895

119905 119909 (119905

0) = 1199090

(23)



119869 (119892) = E [int

1199051

1199050

exp (minus119903119891(119897 minus 1199050)) 119887 (119897 119909

119897 119892 (119897 119909

119897)) 119889119897

+ exp (minus119903119891(1199051minus 1199050)) 1198871(119909 (1199051)) ]

(24)


where 119892 isin G119860 119879 = [119905

0 1199051] and 119887



119887 (119905 119909 119906) = 1198872(1199062) + 1198873(1199091

1199092) (25)

for 1198872 U2rarr R+ and 119887





= 0







119869 (119892) (26)



119869 (119892) (27)



119892isinG119860

119869 (119892) isin G119860 (28)





119869 (119892)


119869 (119892)

(29)

119869 (119892) = E [int

1199051

1199050

exp (minus119903119891(119897 minus 1199050)) [1198872(1199062

119905) + 1198873(119909119905)] d119905

+ exp (minus119903119891(1199051minus 1199050)) 1198871(119909 (1199051)) ]

(30)


3 R+ rarr R+





1198872(1199062) isin 1198622(R)


11986311990621198872(1199062) = minusinfin


11986311990621198872(1199062) = +infin

119863119906211990621198872(1199062) gt 0 forall119906

2isin R

(31)


2



0 = 119863119905V (119905 119909) +

1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909119905)2

]119863119909119909V (119905 119909)

+ 119909 (119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)119863119909V (119905 119909)

+ [1199061

119905minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ 1199062

119905

lowast

119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050)) 1198872(1199062

119905

lowast

)


[119863119909V (119905 119909)]

2

2119863119909119909V (119905 119909)

119903119910

119905

119879

minus1

119905119910

119905

(32)

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050)) 1198871(119909) (33)


0 = 119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))11986311990621198872(1199062

119905) (34)


1198922lowast

(119905 119909) = 1199062lowast

1198922lowast


with 1199062lowast


lowast= minus

119863119909V (119905 119909)

119909119863119909119909V (119905 119909)

minus1

119905119910

119905

1198923119896lowast



(36)



119869lowast= 119869 (119892

lowast) = E [V (119905 119909

0)] (37)



12(119879timesX)

is given by

0 = 119863119905119908 (119905 119909)


[1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2

+(119909)2119879119905]119863119909119909119908 (119905 119909)

+ [ (119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

+119903119910

119905

119879

) 119909]119863119909119908 (119905 119909)

+ [1199061

119905+ 1199062minus 119903119890

119905minus (120590119890)2

]119863119909119908 (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) [1198872(1199062) + 1198873(119909)] ]

119908 (1199051 119909) = exp (minus119903

119891(1199051minus 1199050)) 1198871(119909)

(38)



(b) depending on




inf1199062isinR

1198672(119905 119909 119906

2)

1198672(119905 119909 119906) = 119906

2119863119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) 1198872(1199062)

11986311990621198672(119905 119909 119906) = 119863

119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050))11986311990621198872(1199062) = 0

(39)


2lowast

isin U = R

and

inf1199062isinR

1198672(119905 119909 119906

2) = 119867

2(119905 119909 119906

2lowast

)

= 1199062lowast

119863119909V (119905 119909)


)

(40)




infisinR119896

1198673(119905 119909 )

1198673(119905 119909 ) =

1

2[119879

119905119905119905] (119909)2119863119909119909V (119905 119909)

+119903119910

119905

119879

119909119863119909V (119905 119909)

=1

2(119905+ (

119909 [119863119909V (119905 119909)]

(119909)2119863119909119909V (119905 119909)

) minus1

119905119910

119905)

119879

times 119905(119909)2119863119909119909V (119905 119909)

times (119905+ (

119909 [119863119909V (119905 119909)]

(119909)2119863119909119909V (119905 119909)

) minus1

119905119910

119905)

minus1

2(

[119909119863119909V (119905 119909)]

2

(119909)2119863119909119909V (119905 119909)

)119903119910

119905

119879

minus1

119905119910

119905

119905(119909)2119863119909119909V (119905 119909) gt 0 forall119909 isin R 0

(41)

Hence we have that

lowast= minus

119863119909V (119905 119909)

119909119863119909119909V (119905 119909)

minus1

119905119910

119905

1198923119896lowast

(119905 119909) =

119896lowast if

119896

sum

119894=1

119894lowast

isin [0 1]

119896lowast

sum119896

119895=1119895lowast

if119896

sum

119894=1

119894lowast

gt 1


1198673(119905 119909

lowast) = minus

[119863119909V (119905 119909)]

2

2119863119909119909V (119905 119909)

(119910

119905)119879

minus1

119905119910

119905

(42)





0 = 119863119905V (119905 119909)


1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2

+(119909)2(119879119905) ]119863

119909119909V (119905 119909)

+ [119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+(120590119894)2

+119903119910

119905

119879

] 119909

+ [1199061

119905+ 1199062minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) [1198872(1199062) + 1198873(119909)]

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050)) 1198871(119909)

(43)



lowast=

119869(119892lowast) = E[V(119905

0 1199090)]



119869 (119892) = E [int

1199051

1199050

exp (minus119903119891(119897 minus 1199050))

times [1

21198882((1199062)2

(119897)) +1

21198883(119909119905minus 119897119903)2

] 119889119897

+1

21198881(119909 (1199051) minus 119897119903)2 exp (minus119903

119891(1199051minus 1199050)) ]

(44)


1198871(119909) =

1

21198881(119909 minus 119897

119903)2


1198872(1199062) =

1

21198882(1199062)2


(45)

1198873(119909) =

1

21198883(119909 minus 119897

119903)2

1198883isin (0infin) (46)


minus119905= minus

(119902119905)2

1198882

+ 1198883+ 1199021199052 (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+ 119902119905[minus119903119891minus119903119910

119905

119879

minus1

119905119910

119905+ (120590119890)2

+ (120590119894)2

] 1199021199051= 1198881

(47)

minus 119903

119905= minus

1198883(119909119903

119905minus 119897119903)

119902119905

minus 119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

minus [1199061

119905minus 119903119890

119905minus (120590119890)2

] minus (119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

)

minus (120590119894)2

119909119903(1199051) = 119897119903

(48)

minus 119897119905= minus119903119891119897119905+ 1198883(119909119903

119905minus 119897119903)2

minus 119902119905(120590119890)2

(119909119903

119905minus 1)2

minus 119902119905(120590119894)2

(119909119903

119905)2

119897 (1199051) = 0

(49)





1199062lowast

119905= minus

(119909 minus 119909119903

119905) 119902119905

1198882

1198922lowast

(119905 119909) = 1199062lowast

1198922lowast


lowast

119905= minus

(119909 minus 119909119903

119905)

119909minus1

119905119910

119905 1198923lowast

119879 timesX 997888rarr R119896

(50)

1198923119896lowast




(51)


V (119905 119909) = exp (minus119903119891(1199051minus 1199050)) [

1

2(119909 minus 119909

119903

119905)2

119902119905+

1

2119897119905] (52)

119869lowast= 119869 (119892

lowast) = E [V (119905

0 1199090)] (53)




minus119905= minus119887(119902

119905)2

+ 119888 + 2119886119905119902119905 1199021199051= 1198881

119887 =1

1198882 119888 = 119888

3isin (0infin) 119886 119879 997888rarr R

(54)


1199021

1199051= 119902 (119905

1minus 119905) 119886

1

119905= 119886 (119905

1minus 119905) 119902

1 [0 1199051minus 1199050] 997888rarr R

1

119905= minus (119905

1minus 119905) = minus119887(119902

1

119905)2

+ 119888 + 21198861

1199051199021

119905 1199021

1199050= 1198881

0 = 1

119905+ 119887(1199021

119905)2

minus 119888 minus 21198861

119905119902119905 1199021

0= 1198881

(55)


minus2

119905= minus119887(119902

2

119905)2

+ 119888 1199022

1199051= 1198881 119887 119888 isin (0infin) (56)


119905gt 0 for all 119905 isin 119879 = [0 119905

1minus 1199050]


1minus 1199050] 1199021119905

ge

1199022

119905gt 0 if


0 119887) ge (


119905

minus1198861

119905119887


(57)



(119886119905)2


1

1198882minus

1

1198882)

(58)


=


1minus 1199050]




1

1198882minus

1

1198882ge 0

(119886119905)2

le 1198884le (119888 minus 119888) (119887 minus 119887) = (119888

3minus 1198883) (

1

1198882minus

1

1198882)

(59)


1minus 1199050] 119902(1199051minus

119905) = 1199021

119905ge 1199022

119905gt 0




Note first that

V (119905 119909) = exp (minus119903119891(119905 minus 1199050)) [

1

2(119909 minus 119909

119903

119905)2

119902119905+

1

2119897119905]

119863119905V (119905 119909) = minus119903

119891V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))


119905) 119903

119905119902119905+

1

2(119909 minus 119909

119903

119905)2

119905+

1

2

119897119905]

119863119909V (119905 119909) = exp (minus119903

119891(119905 minus 1199050)) (119909 minus 119909

119903

119905) 119902119905

119863119909119909V (119905 119909) = exp (minus119903

119891(119905 minus 1199050)) 119902119905

(60)


So

0 = 119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))11986311990621198872(1199062)


119903

119905) 119902119905+ 11988821199062]

1199062lowast

= minus(119909 minus 119909

119903

119905) 119902119905

1198882

lowast= minus

(119909 minus 119909119903

119905) 119902119905

119909119902119905

minus1

119905119910

119905

(61)


1199021199051= 1198881 119909

119903(1199051) = 119897119903 119897 (119905

1) = 0

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050))

times [1

2(119909 minus 119909

119903(1199051))2

1199021199051+

1

2119897 (1199051)]

= exp (minus119903119891(1199051minus 1199050))

1

21198881(119909 minus 119897

119903)2

= exp (minus119903119891(1199051minus 1199050)) 1198871(119909)


exp (+119903119891(119905 minus 1199050))

times [119863119905V (119905 119909) +

1

2[(120590119890)2

(1 minus 119909)2

+(120590119894)2

(119909)2]119863119909119909V (119905 119909)

+ 119909 [119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]119863119909V (119905 119909)

+ [1199061

119905minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ 1199062lowast

119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050)) 1198872(1199062lowast

)

+ exp (minus119903119891(119905 minus 1199050)) 1198873(119909)

minus1

2

(119863119909V (119905 119909))

2

119863119909119909V (119905 119909)

119903119910

119905

119879

minus1

119905119910

119905]

= minus119903119891 1

2(119909 minus 119909

119903

119905)2

119902119905minus

1

2119903119891119897119905minus (119909 minus 119909

119903

119905) 119902119905119903

119905

+1

2(119909 minus 119909

119903

119905)2

119905+

1

2

119897119905

+1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2] 119902119905

+ (119909 minus 119909119903

119905) 119902119905[119909 (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+1199061

119905minus 119903119890

119905minus (120590119890)2

]

minus1

2

(119909 minus 119909119903

119905)2

(119902119905)2

1198882

+1

21198883(119909 minus 119897

119903)2

minus1

2(119909 minus 119909

119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

=1

2(119909)2[ minus 119903119891119902119905+ 119905+ (120590119890)2

119902119905+ (120590119894)2

119902119905

+ 2119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

minus(119902119905)2

1198882

+ 1198883minus 119902119905

119903119910

119905

119879

minus1

119905119910

119905]

+ 119909 [ + 119903119891119909119903

119905119902119905minus 119902119905119903

119905minus 119909119903

119905119905minus (120590119890)2

119902119905

minus 119909119903

119905119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ 119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] +119909119903

119905(119902119905)2

1198882

minus1198883119897119903+ 119909119903

119905119902119905

119903119910

119905

119879

minus1

119905119910

119905]

+1

2(119909)0[ minus 119903119891(119909119903

119905)2

119902119905+ 2119909119903

119905119902119905119903

119905+ (119909119903

119905)2

119905minus 119903119891119897119905

+ (120590119890)2

119902119905minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

]

minus(119909119903

119905)2

(119902119905)2

1198882

+ 1198883(119897119903)2

minus(119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905+ 119897119905]

(62)



minus 119902119905119903

119905minus 119909119903

119905119905+ 119903119891119909119903

119905119902119905minus 2(120590119890)2

119902119905

minus 119909119903

119905119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ 119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

]

+119909119903

119905(119902119905)2

1198882

minus 1198883119897119903+ 119909119903

119905119902119905

119903119910

119905

119879

minus1

119905119910

119905

= minus119902119905119903

119905

+ 119909119903

119905[minus

(119902119905)2

1198882

+ 1198883

+ 2119902119905(119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+119902119905((120590119890)2

+ (120590119894)2

minus 119903119891minus119903119910

119905

119879

minus1

119905119910

119905)]

minus 119909119903

119905119902119905[ (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

minus119903119891minus119903119910

119905

119879

minus1

119905119910

119905]

+ 119902 [minus(120590119890)2

+ (1199061

119905minus 119903119890

119905minus (120590119890)2

)] +119909119903

119905(119902119905)2

1198882

minus 1198883119897119903

= 119902119905[minus119903

119905+

1198883(119909119903

119905minus 119897119903)

119902119905

]

+ 119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ (119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

) + (120590119894)2

+ (1199061

119905minus 119903119890

119905minus (120590119890)2

)

= 0

(63)



119897119905minus 119903119891119897119905minus 119903119891(119909119903

119905)2

119902119905

+ 2119909119903

119905[119902119905119903

119905+ 119909119903

119905119905] minus (119909

119903

119905)2

119905+ (120590119890)2

119902119905

minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] + 1198883(119897119903)2

minus(119909119903

119905)2

(119902119905)2

1198882

minus (119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905


119897119905minus 119903119891119897119905minus 119903119891(119909119903

119905)2

119902119905+ (120590119890)2

119902119905

minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] + 1198883(119897119903)2

minus(119909119903

119905)2

(119902119905)2

1198882

minus (119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

+ 2119903119891(119909119903

119905)2

119902119905minus 2119909119903

119905119902119905(120590119890)2

minus 2(119909119903

119905)2

119902119905[119903

T+ 119903119890minus 119903119894+ (120590119890)2

+ (120590119894)2

]

+ 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] minus 11988831198971199032119909119903

119905

+2(119909119903

119905)2

(119902119905)2

1198882

+ 2(119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

minus(119909119903

119905)2

(119902119905)2

1198882

+ 1198883(119909119903

119905)2

+ (119909119903

119905)2

1199021199052 (119903

T+ 119903119890minus 119903119894+ (120590119890)2

+ (120590119894)2

)

+ (119909119903

119905)2

119902119905(minus119903119891+ (120590119890)2

+ (120590119894)2

minus119903119910

119905

119879

minus1

119905119910

119905)

= 119897119905minus 119903119891119897119905+ 1198883(119909119903

119905minus 119897119903)2

minus (120590119890)2

119902119905(119909119903

119905minus 1)2

minus (120590119894)2

119902119905(119909119903

119905)2

= 0

(64)







2gt 0 should


2(1199062) = (12)119888

2(1199062)2 given by (45) This







3(119909) = (12)119888



1198873(119909) = 119888

3[exp (119909 minus 119897

119903) + (119897119903minus 119909) minus 1] (65)







1199051198882 which





119903(119909) = (12)119888

3(119909 minus 119909




119903



119905119910



minus1

119905119910


deviation (119909 minus 119909119903

119905)




1199061

119905minus 119903119890

119905minus (120590119890)2

(66)



119890

119905is the


119903119890



1199051 119909119905lt 119897119903The term 119888

3(119909119903





119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

] (67)




T(119905) + (120590

119890)2+ (120590119894)2 is the effect of


(119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

) minus (120590119894)2

(68)




minus119903

119905= minus119896 (119909

119903

119905minus 119898) 119909

119903(1199051) = 119897119903

119896 = (119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2

+ (120590119894)2

)) +1198883

1199020

119898 =


) + (120590119890)2

(119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2+ (120590119894)2

)) + 11988831199020

(69)


119902119905

= 1199020 and lim

119905darr0119909119903




1199062lowast

119905= minus

(119909119905minus 119898) 119902

0

1198882

= gt 0 if 119909

119905lt 119898

lt 0 if 119909119905gt 119898

120587lowast

119905= minus

(119909119905minus 119898)

119909119905

119895

= gt 0 if 119909

119905lt 119898


(70)


Case 1 (119909119905











119898 =


) + (120590119890)2

(119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2+ (120590119894)2

)) + 11988831199020

(71)





Appendices













References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





119896

119905 119896 isin N

119899=


119910 = (T (119905) 1199101

119905 119910

119899

119905) (8)


119905 119910

119899


model 119910 as

119889119910119905= 119903119910

119905119889119905 + Σ

119910

119905119889119882119910

119905 119910 (119905

0) = 1199100 (9)





119878 = 120587 isin R119899+1

120587 = (1205870 120587

119899)119879

1205870+ sdot sdot sdot + 120587

119899= 1

1205870ge 0 120587

119899ge 0

(10)


119889ℎ119905= 120587119879

119905119889119910119905= 120587119879

119905119903119910

119905119889119905 + 120587

119879

119905Σ119910

119905119889119882119910

119905 (11)


119903T(119905) = 119903

1199100

(119905) 119903T 119879 997888rarr R

+


119903119910

119905= (119903

T(119905)

119903119910

119905

119879

+ 119903T(119905) 1119899)

119879

119910 119879 997888rarr R

119899

120587119905= (1205870

119905 119879

119905)119879

= (1205870

119905 1205871

119905 120587

119896

119905)119879

119879 997888rarr R119896

Σ119910

119905= (

0 sdot sdot sdot 0

Σ119910

119905

) Σ119910

119905isin R119899times119899

119905= Σ119910

119905

Σ119910

119905

119879

Thenwe have that

120587119879

119905119903119910

119905= 1205870

119905119903T(119905) +

119895119879

119905119910

119905+ 119895119879

119905119903T(119905) 1119899

= 119903T(119905) +

119879

119905119910

119905

120587119879

119905Σ119910

119905119889119882119910

119905= 119879

119905Σ119910

119905119889119882119910

119905

119889ℎ119905= [119903

T(119905) +

119879

119905119910

119905] 119889119905 +

119879

119905Σ119910

119905119889119882119910

119905 ℎ (119905

0) = ℎ0

(12)






119889119894119905= 119903119894

119905119889119905 + 120590

119894119889119882119894

119905 119894 (119905

0) = 1198940 (13)




1198891199091

119905= 1199091

119905119889ℎ119905+ 1199092

1199051199063

119905119889119905 minus 119909

2

119905119889119890119905

= [119903T(119905) 1199091

119905+ 1199091

119905119879

119905119910

119905+ 1199092

1199051199061

119905+ 1199092

1199051199062

119905minus 1199092

119905119903119890

119905] 119889119905

+ [1199091

119905119879

119905Σ119910

119905119889119882119910

119905minus 1199092

119905120590119890119889119882119890

119905]

1198891199092

119905= 1199092

119905119889119894119905minus 1199092

119905119889119890119905

= 1199092

119905[119903119894

119905119889119905 + 120590

119894119889119882119894

119905] minus 1199092

119905[119903119890

119905119889119905 + 120590

119890119889119882119890

119905]

= 1199092

119905[119903119894

119905minus 119903119890

119905] 119889119905 + 119909

2

119905[120590119894119889119882119894

119905minus 120590119890119889119882119890

119905]

(14)



119889119909119905= 119860119905119909119905119889119905 + 119873 (119909

119905) 119906119905119889119905 + 119886

119905119889119905 + 119878 (119909

119905 119906119905) 119889119882119905 (15)


119906119905= [

1199062

119905

119905

] 119906 Ω times 119879 997888rarr R119899+1

119860119905= [

119903T(119905) minus119903

119890

119905

0 119903119894

119905minus 119903119890

119905

]

119873 (119909119905) = [

1199092

1199051199091

119905

119903119910

119905

119879

0 0

] 119886119905= [

1199092

1199051199061

119905

0]

119878 (119909119905 119906119905) = [

1199091

119905119879

119905Σ119910

119905minus1199092

119905120590119890

0

0 minus1199092

119905120590119890

1199092

119905120590119894]

119882119905= [

[

119882119910

119905

119882119890

119905

119882119894

119905

]

]

(16)


where 119882119910

119905 119882119890119905 and 119882

119894



119905gt 0 120590119894

119905gt 0 and




119873(119909119905) 119906119905= [

1199092

119905

0] 1199062

119905+ [

1199091

119905

119903119910

119905

119879

0

] 119905

= [0 1

0 0] 1199091199051199063

119905+

119899

sum

119898=1

[1199091

119905119910119898

119905

0] 119898

119905

= 11986101199091199051199060

119905+

119899

sum

119898=1

[119910119898

1199050

0 0] 119909119905119898

119905

=

119899

sum

119898=0

[119861119898119909119905] 119906119898

119905

119878 (119909119905 119906119905) 119889119882119905= [

[119879

119905119905119905]12

0

0 0

] 1199091199051198891198821

119905

+ [0 minus120590119890

0 minus120590119890] 1199091199051198891198822

119905+ [

0 0

0 120590119894] 1199091199051198891198823

119905

=

3

sum

119895=1

[119872119895119895(119906119905) 119909119905] 119889119882119895119895

119905

(17)






of 1199091119905via the term 119909

1

119905

119903119910

119905

119879



119889119909119905= 119860119905119909119905119889119905 +

119899

sum

119895=0

[119861119895119909119905] 119906119895

119905119889119905 + 119886

119905119889119905

+

3

sum

119895=1

[119872119895(119906119905) 119909119905] 119889119882119895119895

119905

(18)


119905=

1199091

1199051199092

119905


119889119909119905= 119909119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

+119903119910

119905

119879

119905] 119889119905

+ [1199061

119905+ 1199062

119905minus 119903119890

119905minus (120590119890)2

] 119889119905

+ [(120590119890)2

(1 minus 119909119905)2

+ (120590119894)2

1199092

119905+ 1199092

119905119879

119905119905119905]

12

119889119882119905

119909 (1199050) = 1199090

(19)


1 1199092)119879 as in (14) Next we

determine

119889(1199092

119905)minus1

= minus(1199092

119905)minus2

1198891199092

119905+

1

22(1199092)minus3

119905119889 lt 119909

2 1199092gt 119905


119905(119903119894

119905minus 119903119890

119905) + (119909

2

119905)minus1

((120590119890)2

+ (120590119894)2

)] 119889119905

minus (1199092

119905)minus1

[0 minus120590119890

120590119894] 119889119882119905

119889119909119905= 1199091

119905119889(1199092

119905)minus1

+ (1199092

119905)minus1

1198891199091

119905+ 119889 lt 119909

1 (119909

2)minus1

gt 119905

= [119903T(119905) 119909119905minus 119903119890

119905+ 1199061

119905+ 1199062

119905+ 119909119905119910

119905119905

minus119909119905[119903119894

119905minus 119903119890

119905] + 119909119905((120590119890)2

+ (120590119894)2

) minus (120590119890)2

] 119889119905

+ (119909119905119879

119905Σ119910

119905minus 120590119890(1 minus 119909

119905) minus 120590119894119909119905) 119889119882119905

= 119909119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

+119903119910

119905

119879

119905] 119889119905

+ [1199061

119905+ 1199062

119905minus 119903119890

119905minus (120590119890)2

] 119889119905

+ [(120590119890)2

(1 minus 119909119905)2

+ (120590119894)2

(119909119905)2

+(119909119905)2

119879

119905119905119905+ (120590119894)2

(119909119905)2

]

12

119889119882119905

(20)


minus119903119890

119905+ 119909119905119903119890

119905= minus119903119890

119905(119909119905minus 1) (21)


119890)2+ 119909119905(120590119890)2= (120590119890)2(119909119905minus 1) appears




Time (years)


0 01 02 03 04 05 06 07 08 09 1

5

0

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

minus45



Parameter Value119862 500119903119894 001119910 0061199062 003

119903119862 005120590119890 16

04 150119903119890 004120590119894 18

1199061 001

119882 003











119905in (4)















(22)


Table4Summaryof

netstablefun

ding

ratio

Availables

tablefun

ding

(sou

rces)

Requ

iredstablefund

ing(uses)

Item

Avlblty

Fctr

Item

Rqrd

Fctr

(i)T1KandT2

Kinstr

uments

(ii)O

ther

preferredshares

andcapitalinstrum

entsin

excessof

T2Kallowableam

ount

having

aneffectiv

ematurity

of1Y

rorgt

1Yr

(iii)Other

liabilitiesw

ithan

effectiv

ematurity

of1Y

rorgt

1Yr

100

(i)Ca

sh(ii)S

hort-te

rmun

securedactiv

elytraded

instr

uments(lt1Y

r)(iii)Securitiesw

ithexactly

offsetting

reverser

epo

(iv)S

ecurities

with

remaining

maturity

lt1Y

r(v)N


ancialsw

ithremaining

maturity

lt1Y

r

0

Stabledepo

sitso

fretailand

smallbusinessc

ustomers

(non

maturity

orresid

ualm

aturity

lt1Y

r)90

Debtissuedor

guaranteed


sBISIM

FEC

non

centralgovernm


entb

anks

with

a0ris

kweightu

nder

BaselIIS

tand

ardizedAp

proach

5

Lessstabledepo

sitso

fretailand

smallbusinessc

ustomers

(non

maturity

orresid

ualm

aturity

lt1Y

r)80

Unencum

beredno

nfinancialsenioru

nsecured

corporateb

onds

and

coveredbo

ndsrated

atleastA

Aminusanddebt

thatisissuedby

SovereignsC

entralBa

nksandPS

Eswith

arisk

weightin

gof

20

maturity

ge1Y

r

20

Who

lesalefund

ingprovided

byno

nfinancialcorpo

ratecusto

mers


ultilateraldevelopm

entb

anksand

PSEs

(non

maturity

orresid

ualm

aturity

lt1Y

r)50

(i)Unencum


rnon

financialsenior

unsecuredcorporateb

onds

(orc

overed

bond

s)ratedfro

mA+to

Aminus

maturity

ge1Y

r(ii)G

old

(iii)Lo

anstono

nfinancialcorpo

rateclients

SovereignsC

entral

Bank

sandPS

Eswith

amaturity

lt1yr

50


ndequityno

tincludedabove

0

Unencum

beredresid

entia

lmortgages

ofanymaturity

andother


ding

loanstofin

ancialinstitu

tions

with

aremaining

maturity

of1Y

rorgt

1Yrthatw

ould

qualify

forthe

35or

lower

riskweightu

nder

baselIIstand

ardizedapproach

forc

redit

risk

65

Other


allbusinessesh

avingam

aturity

lt1Y

r85

Allothera

ssets

100

Offbalances

heetexpo

sures

Und

rawnam

ount

ofcommitted

creditandliq

uidityfacilities

5

Other

contingent

fund

ingob

ligations

National

superviso

rydiscretio

n






5



50











119889119909119905= 119860119905119909119905119889119905 +

119899

sum

119895=0

119861119895119909119905119892119895(119905 119909119905) 119889119905 + 119886

119905119889119905

+

3

sum

119895=1

119872119895119895(119892 (119905 119909

119905)) 119909119905119889119882119895119895

119905 119909 (119905

0) = 1199090

(23)



119869 (119892) = E [int

1199051

1199050

exp (minus119903119891(119897 minus 1199050)) 119887 (119897 119909

119897 119892 (119897 119909

119897)) 119889119897

+ exp (minus119903119891(1199051minus 1199050)) 1198871(119909 (1199051)) ]

(24)


where 119892 isin G119860 119879 = [119905

0 1199051] and 119887



119887 (119905 119909 119906) = 1198872(1199062) + 1198873(1199091

1199092) (25)

for 1198872 U2rarr R+ and 119887





= 0







119869 (119892) (26)



119869 (119892) (27)



119892isinG119860

119869 (119892) isin G119860 (28)





119869 (119892)


119869 (119892)

(29)

119869 (119892) = E [int

1199051

1199050

exp (minus119903119891(119897 minus 1199050)) [1198872(1199062

119905) + 1198873(119909119905)] d119905

+ exp (minus119903119891(1199051minus 1199050)) 1198871(119909 (1199051)) ]

(30)


3 R+ rarr R+





1198872(1199062) isin 1198622(R)


11986311990621198872(1199062) = minusinfin


11986311990621198872(1199062) = +infin

119863119906211990621198872(1199062) gt 0 forall119906

2isin R

(31)


2



0 = 119863119905V (119905 119909) +

1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909119905)2

]119863119909119909V (119905 119909)

+ 119909 (119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)119863119909V (119905 119909)

+ [1199061

119905minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ 1199062

119905

lowast

119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050)) 1198872(1199062

119905

lowast

)


[119863119909V (119905 119909)]

2

2119863119909119909V (119905 119909)

119903119910

119905

119879

minus1

119905119910

119905

(32)

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050)) 1198871(119909) (33)


0 = 119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))11986311990621198872(1199062

119905) (34)


1198922lowast

(119905 119909) = 1199062lowast

1198922lowast


with 1199062lowast


lowast= minus

119863119909V (119905 119909)

119909119863119909119909V (119905 119909)

minus1

119905119910

119905

1198923119896lowast



(36)



119869lowast= 119869 (119892

lowast) = E [V (119905 119909

0)] (37)



12(119879timesX)

is given by

0 = 119863119905119908 (119905 119909)


[1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2

+(119909)2119879119905]119863119909119909119908 (119905 119909)

+ [ (119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

+119903119910

119905

119879

) 119909]119863119909119908 (119905 119909)

+ [1199061

119905+ 1199062minus 119903119890

119905minus (120590119890)2

]119863119909119908 (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) [1198872(1199062) + 1198873(119909)] ]

119908 (1199051 119909) = exp (minus119903

119891(1199051minus 1199050)) 1198871(119909)

(38)



(b) depending on




inf1199062isinR

1198672(119905 119909 119906

2)

1198672(119905 119909 119906) = 119906

2119863119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) 1198872(1199062)

11986311990621198672(119905 119909 119906) = 119863

119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050))11986311990621198872(1199062) = 0

(39)


2lowast

isin U = R

and

inf1199062isinR

1198672(119905 119909 119906

2) = 119867

2(119905 119909 119906

2lowast

)

= 1199062lowast

119863119909V (119905 119909)


)

(40)




infisinR119896

1198673(119905 119909 )

1198673(119905 119909 ) =

1

2[119879

119905119905119905] (119909)2119863119909119909V (119905 119909)

+119903119910

119905

119879

119909119863119909V (119905 119909)

=1

2(119905+ (

119909 [119863119909V (119905 119909)]

(119909)2119863119909119909V (119905 119909)

) minus1

119905119910

119905)

119879

times 119905(119909)2119863119909119909V (119905 119909)

times (119905+ (

119909 [119863119909V (119905 119909)]

(119909)2119863119909119909V (119905 119909)

) minus1

119905119910

119905)

minus1

2(

[119909119863119909V (119905 119909)]

2

(119909)2119863119909119909V (119905 119909)

)119903119910

119905

119879

minus1

119905119910

119905

119905(119909)2119863119909119909V (119905 119909) gt 0 forall119909 isin R 0

(41)

Hence we have that

lowast= minus

119863119909V (119905 119909)

119909119863119909119909V (119905 119909)

minus1

119905119910

119905

1198923119896lowast

(119905 119909) =

119896lowast if

119896

sum

119894=1

119894lowast

isin [0 1]

119896lowast

sum119896

119895=1119895lowast

if119896

sum

119894=1

119894lowast

gt 1


1198673(119905 119909

lowast) = minus

[119863119909V (119905 119909)]

2

2119863119909119909V (119905 119909)

(119910

119905)119879

minus1

119905119910

119905

(42)





0 = 119863119905V (119905 119909)


1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2

+(119909)2(119879119905) ]119863

119909119909V (119905 119909)

+ [119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+(120590119894)2

+119903119910

119905

119879

] 119909

+ [1199061

119905+ 1199062minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) [1198872(1199062) + 1198873(119909)]

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050)) 1198871(119909)

(43)



lowast=

119869(119892lowast) = E[V(119905

0 1199090)]



119869 (119892) = E [int

1199051

1199050

exp (minus119903119891(119897 minus 1199050))

times [1

21198882((1199062)2

(119897)) +1

21198883(119909119905minus 119897119903)2

] 119889119897

+1

21198881(119909 (1199051) minus 119897119903)2 exp (minus119903

119891(1199051minus 1199050)) ]

(44)


1198871(119909) =

1

21198881(119909 minus 119897

119903)2


1198872(1199062) =

1

21198882(1199062)2


(45)

1198873(119909) =

1

21198883(119909 minus 119897

119903)2

1198883isin (0infin) (46)


minus119905= minus

(119902119905)2

1198882

+ 1198883+ 1199021199052 (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+ 119902119905[minus119903119891minus119903119910

119905

119879

minus1

119905119910

119905+ (120590119890)2

+ (120590119894)2

] 1199021199051= 1198881

(47)

minus 119903

119905= minus

1198883(119909119903

119905minus 119897119903)

119902119905

minus 119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

minus [1199061

119905minus 119903119890

119905minus (120590119890)2

] minus (119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

)

minus (120590119894)2

119909119903(1199051) = 119897119903

(48)

minus 119897119905= minus119903119891119897119905+ 1198883(119909119903

119905minus 119897119903)2

minus 119902119905(120590119890)2

(119909119903

119905minus 1)2

minus 119902119905(120590119894)2

(119909119903

119905)2

119897 (1199051) = 0

(49)





1199062lowast

119905= minus

(119909 minus 119909119903

119905) 119902119905

1198882

1198922lowast

(119905 119909) = 1199062lowast

1198922lowast


lowast

119905= minus

(119909 minus 119909119903

119905)

119909minus1

119905119910

119905 1198923lowast

119879 timesX 997888rarr R119896

(50)

1198923119896lowast




(51)


V (119905 119909) = exp (minus119903119891(1199051minus 1199050)) [

1

2(119909 minus 119909

119903

119905)2

119902119905+

1

2119897119905] (52)

119869lowast= 119869 (119892

lowast) = E [V (119905

0 1199090)] (53)




minus119905= minus119887(119902

119905)2

+ 119888 + 2119886119905119902119905 1199021199051= 1198881

119887 =1

1198882 119888 = 119888

3isin (0infin) 119886 119879 997888rarr R

(54)


1199021

1199051= 119902 (119905

1minus 119905) 119886

1

119905= 119886 (119905

1minus 119905) 119902

1 [0 1199051minus 1199050] 997888rarr R

1

119905= minus (119905

1minus 119905) = minus119887(119902

1

119905)2

+ 119888 + 21198861

1199051199021

119905 1199021

1199050= 1198881

0 = 1

119905+ 119887(1199021

119905)2

minus 119888 minus 21198861

119905119902119905 1199021

0= 1198881

(55)


minus2

119905= minus119887(119902

2

119905)2

+ 119888 1199022

1199051= 1198881 119887 119888 isin (0infin) (56)


119905gt 0 for all 119905 isin 119879 = [0 119905

1minus 1199050]


1minus 1199050] 1199021119905

ge

1199022

119905gt 0 if


0 119887) ge (


119905

minus1198861

119905119887


(57)



(119886119905)2


1

1198882minus

1

1198882)

(58)


=


1minus 1199050]




1

1198882minus

1

1198882ge 0

(119886119905)2

le 1198884le (119888 minus 119888) (119887 minus 119887) = (119888

3minus 1198883) (

1

1198882minus

1

1198882)

(59)


1minus 1199050] 119902(1199051minus

119905) = 1199021

119905ge 1199022

119905gt 0




Note first that

V (119905 119909) = exp (minus119903119891(119905 minus 1199050)) [

1

2(119909 minus 119909

119903

119905)2

119902119905+

1

2119897119905]

119863119905V (119905 119909) = minus119903

119891V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))


119905) 119903

119905119902119905+

1

2(119909 minus 119909

119903

119905)2

119905+

1

2

119897119905]

119863119909V (119905 119909) = exp (minus119903

119891(119905 minus 1199050)) (119909 minus 119909

119903

119905) 119902119905

119863119909119909V (119905 119909) = exp (minus119903

119891(119905 minus 1199050)) 119902119905

(60)


So

0 = 119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))11986311990621198872(1199062)


119903

119905) 119902119905+ 11988821199062]

1199062lowast

= minus(119909 minus 119909

119903

119905) 119902119905

1198882

lowast= minus

(119909 minus 119909119903

119905) 119902119905

119909119902119905

minus1

119905119910

119905

(61)


1199021199051= 1198881 119909

119903(1199051) = 119897119903 119897 (119905

1) = 0

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050))

times [1

2(119909 minus 119909

119903(1199051))2

1199021199051+

1

2119897 (1199051)]

= exp (minus119903119891(1199051minus 1199050))

1

21198881(119909 minus 119897

119903)2

= exp (minus119903119891(1199051minus 1199050)) 1198871(119909)


exp (+119903119891(119905 minus 1199050))

times [119863119905V (119905 119909) +

1

2[(120590119890)2

(1 minus 119909)2

+(120590119894)2

(119909)2]119863119909119909V (119905 119909)

+ 119909 [119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]119863119909V (119905 119909)

+ [1199061

119905minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ 1199062lowast

119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050)) 1198872(1199062lowast

)

+ exp (minus119903119891(119905 minus 1199050)) 1198873(119909)

minus1

2

(119863119909V (119905 119909))

2

119863119909119909V (119905 119909)

119903119910

119905

119879

minus1

119905119910

119905]

= minus119903119891 1

2(119909 minus 119909

119903

119905)2

119902119905minus

1

2119903119891119897119905minus (119909 minus 119909

119903

119905) 119902119905119903

119905

+1

2(119909 minus 119909

119903

119905)2

119905+

1

2

119897119905

+1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2] 119902119905

+ (119909 minus 119909119903

119905) 119902119905[119909 (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+1199061

119905minus 119903119890

119905minus (120590119890)2

]

minus1

2

(119909 minus 119909119903

119905)2

(119902119905)2

1198882

+1

21198883(119909 minus 119897

119903)2

minus1

2(119909 minus 119909

119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

=1

2(119909)2[ minus 119903119891119902119905+ 119905+ (120590119890)2

119902119905+ (120590119894)2

119902119905

+ 2119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

minus(119902119905)2

1198882

+ 1198883minus 119902119905

119903119910

119905

119879

minus1

119905119910

119905]

+ 119909 [ + 119903119891119909119903

119905119902119905minus 119902119905119903

119905minus 119909119903

119905119905minus (120590119890)2

119902119905

minus 119909119903

119905119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ 119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] +119909119903

119905(119902119905)2

1198882

minus1198883119897119903+ 119909119903

119905119902119905

119903119910

119905

119879

minus1

119905119910

119905]

+1

2(119909)0[ minus 119903119891(119909119903

119905)2

119902119905+ 2119909119903

119905119902119905119903

119905+ (119909119903

119905)2

119905minus 119903119891119897119905

+ (120590119890)2

119902119905minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

]

minus(119909119903

119905)2

(119902119905)2

1198882

+ 1198883(119897119903)2

minus(119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905+ 119897119905]

(62)



minus 119902119905119903

119905minus 119909119903

119905119905+ 119903119891119909119903

119905119902119905minus 2(120590119890)2

119902119905

minus 119909119903

119905119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ 119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

]

+119909119903

119905(119902119905)2

1198882

minus 1198883119897119903+ 119909119903

119905119902119905

119903119910

119905

119879

minus1

119905119910

119905

= minus119902119905119903

119905

+ 119909119903

119905[minus

(119902119905)2

1198882

+ 1198883

+ 2119902119905(119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+119902119905((120590119890)2

+ (120590119894)2

minus 119903119891minus119903119910

119905

119879

minus1

119905119910

119905)]

minus 119909119903

119905119902119905[ (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

minus119903119891minus119903119910

119905

119879

minus1

119905119910

119905]

+ 119902 [minus(120590119890)2

+ (1199061

119905minus 119903119890

119905minus (120590119890)2

)] +119909119903

119905(119902119905)2

1198882

minus 1198883119897119903

= 119902119905[minus119903

119905+

1198883(119909119903

119905minus 119897119903)

119902119905

]

+ 119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ (119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

) + (120590119894)2

+ (1199061

119905minus 119903119890

119905minus (120590119890)2

)

= 0

(63)



119897119905minus 119903119891119897119905minus 119903119891(119909119903

119905)2

119902119905

+ 2119909119903

119905[119902119905119903

119905+ 119909119903

119905119905] minus (119909

119903

119905)2

119905+ (120590119890)2

119902119905

minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] + 1198883(119897119903)2

minus(119909119903

119905)2

(119902119905)2

1198882

minus (119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905


119897119905minus 119903119891119897119905minus 119903119891(119909119903

119905)2

119902119905+ (120590119890)2

119902119905

minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] + 1198883(119897119903)2

minus(119909119903

119905)2

(119902119905)2

1198882

minus (119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

+ 2119903119891(119909119903

119905)2

119902119905minus 2119909119903

119905119902119905(120590119890)2

minus 2(119909119903

119905)2

119902119905[119903

T+ 119903119890minus 119903119894+ (120590119890)2

+ (120590119894)2

]

+ 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] minus 11988831198971199032119909119903

119905

+2(119909119903

119905)2

(119902119905)2

1198882

+ 2(119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

minus(119909119903

119905)2

(119902119905)2

1198882

+ 1198883(119909119903

119905)2

+ (119909119903

119905)2

1199021199052 (119903

T+ 119903119890minus 119903119894+ (120590119890)2

+ (120590119894)2

)

+ (119909119903

119905)2

119902119905(minus119903119891+ (120590119890)2

+ (120590119894)2

minus119903119910

119905

119879

minus1

119905119910

119905)

= 119897119905minus 119903119891119897119905+ 1198883(119909119903

119905minus 119897119903)2

minus (120590119890)2

119902119905(119909119903

119905minus 1)2

minus (120590119894)2

119902119905(119909119903

119905)2

= 0

(64)







2gt 0 should


2(1199062) = (12)119888

2(1199062)2 given by (45) This







3(119909) = (12)119888



1198873(119909) = 119888

3[exp (119909 minus 119897

119903) + (119897119903minus 119909) minus 1] (65)







1199051198882 which





119903(119909) = (12)119888

3(119909 minus 119909




119903



119905119910



minus1

119905119910


deviation (119909 minus 119909119903

119905)




1199061

119905minus 119903119890

119905minus (120590119890)2

(66)



119890

119905is the


119903119890



1199051 119909119905lt 119897119903The term 119888

3(119909119903





119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

] (67)




T(119905) + (120590

119890)2+ (120590119894)2 is the effect of


(119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

) minus (120590119894)2

(68)




minus119903

119905= minus119896 (119909

119903

119905minus 119898) 119909

119903(1199051) = 119897119903

119896 = (119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2

+ (120590119894)2

)) +1198883

1199020

119898 =


) + (120590119890)2

(119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2+ (120590119894)2

)) + 11988831199020

(69)


119902119905

= 1199020 and lim

119905darr0119909119903




1199062lowast

119905= minus

(119909119905minus 119898) 119902

0

1198882

= gt 0 if 119909

119905lt 119898

lt 0 if 119909119905gt 119898

120587lowast

119905= minus

(119909119905minus 119898)

119909119905

119895

= gt 0 if 119909

119905lt 119898


(70)


Case 1 (119909119905











119898 =


) + (120590119890)2

(119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2+ (120590119894)2

)) + 11988831199020

(71)





Appendices













References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




where 119882119910

119905 119882119890119905 and 119882

119894



119905gt 0 120590119894

119905gt 0 and




119873(119909119905) 119906119905= [

1199092

119905

0] 1199062

119905+ [

1199091

119905

119903119910

119905

119879

0

] 119905

= [0 1

0 0] 1199091199051199063

119905+

119899

sum

119898=1

[1199091

119905119910119898

119905

0] 119898

119905

= 11986101199091199051199060

119905+

119899

sum

119898=1

[119910119898

1199050

0 0] 119909119905119898

119905

=

119899

sum

119898=0

[119861119898119909119905] 119906119898

119905

119878 (119909119905 119906119905) 119889119882119905= [

[119879

119905119905119905]12

0

0 0

] 1199091199051198891198821

119905

+ [0 minus120590119890

0 minus120590119890] 1199091199051198891198822

119905+ [

0 0

0 120590119894] 1199091199051198891198823

119905

=

3

sum

119895=1

[119872119895119895(119906119905) 119909119905] 119889119882119895119895

119905

(17)






of 1199091119905via the term 119909

1

119905

119903119910

119905

119879



119889119909119905= 119860119905119909119905119889119905 +

119899

sum

119895=0

[119861119895119909119905] 119906119895

119905119889119905 + 119886

119905119889119905

+

3

sum

119895=1

[119872119895(119906119905) 119909119905] 119889119882119895119895

119905

(18)


119905=

1199091

1199051199092

119905


119889119909119905= 119909119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

+119903119910

119905

119879

119905] 119889119905

+ [1199061

119905+ 1199062

119905minus 119903119890

119905minus (120590119890)2

] 119889119905

+ [(120590119890)2

(1 minus 119909119905)2

+ (120590119894)2

1199092

119905+ 1199092

119905119879

119905119905119905]

12

119889119882119905

119909 (1199050) = 1199090

(19)


1 1199092)119879 as in (14) Next we

determine

119889(1199092

119905)minus1

= minus(1199092

119905)minus2

1198891199092

119905+

1

22(1199092)minus3

119905119889 lt 119909

2 1199092gt 119905


119905(119903119894

119905minus 119903119890

119905) + (119909

2

119905)minus1

((120590119890)2

+ (120590119894)2

)] 119889119905

minus (1199092

119905)minus1

[0 minus120590119890

120590119894] 119889119882119905

119889119909119905= 1199091

119905119889(1199092

119905)minus1

+ (1199092

119905)minus1

1198891199091

119905+ 119889 lt 119909

1 (119909

2)minus1

gt 119905

= [119903T(119905) 119909119905minus 119903119890

119905+ 1199061

119905+ 1199062

119905+ 119909119905119910

119905119905

minus119909119905[119903119894

119905minus 119903119890

119905] + 119909119905((120590119890)2

+ (120590119894)2

) minus (120590119890)2

] 119889119905

+ (119909119905119879

119905Σ119910

119905minus 120590119890(1 minus 119909

119905) minus 120590119894119909119905) 119889119882119905

= 119909119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

+119903119910

119905

119879

119905] 119889119905

+ [1199061

119905+ 1199062

119905minus 119903119890

119905minus (120590119890)2

] 119889119905

+ [(120590119890)2

(1 minus 119909119905)2

+ (120590119894)2

(119909119905)2

+(119909119905)2

119879

119905119905119905+ (120590119894)2

(119909119905)2

]

12

119889119882119905

(20)


minus119903119890

119905+ 119909119905119903119890

119905= minus119903119890

119905(119909119905minus 1) (21)


119890)2+ 119909119905(120590119890)2= (120590119890)2(119909119905minus 1) appears




Time (years)


0 01 02 03 04 05 06 07 08 09 1

5

0

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

minus45



Parameter Value119862 500119903119894 001119910 0061199062 003

119903119862 005120590119890 16

04 150119903119890 004120590119894 18

1199061 001

119882 003











119905in (4)















(22)


Table4Summaryof

netstablefun

ding

ratio

Availables

tablefun

ding

(sou

rces)

Requ

iredstablefund

ing(uses)

Item

Avlblty

Fctr

Item

Rqrd

Fctr

(i)T1KandT2

Kinstr

uments

(ii)O

ther

preferredshares

andcapitalinstrum

entsin

excessof

T2Kallowableam

ount

having

aneffectiv

ematurity

of1Y

rorgt

1Yr

(iii)Other

liabilitiesw

ithan

effectiv

ematurity

of1Y

rorgt

1Yr

100

(i)Ca

sh(ii)S

hort-te

rmun

securedactiv

elytraded

instr

uments(lt1Y

r)(iii)Securitiesw

ithexactly

offsetting

reverser

epo

(iv)S

ecurities

with

remaining

maturity

lt1Y

r(v)N


ancialsw

ithremaining

maturity

lt1Y

r

0

Stabledepo

sitso

fretailand

smallbusinessc

ustomers

(non

maturity

orresid

ualm

aturity

lt1Y

r)90

Debtissuedor

guaranteed


sBISIM

FEC

non

centralgovernm


entb

anks

with

a0ris

kweightu

nder

BaselIIS

tand

ardizedAp

proach

5

Lessstabledepo

sitso

fretailand

smallbusinessc

ustomers

(non

maturity

orresid

ualm

aturity

lt1Y

r)80

Unencum

beredno

nfinancialsenioru

nsecured

corporateb

onds

and

coveredbo

ndsrated

atleastA

Aminusanddebt

thatisissuedby

SovereignsC

entralBa

nksandPS

Eswith

arisk

weightin

gof

20

maturity

ge1Y

r

20

Who

lesalefund

ingprovided

byno

nfinancialcorpo

ratecusto

mers


ultilateraldevelopm

entb

anksand

PSEs

(non

maturity

orresid

ualm

aturity

lt1Y

r)50

(i)Unencum


rnon

financialsenior

unsecuredcorporateb

onds

(orc

overed

bond

s)ratedfro

mA+to

Aminus

maturity

ge1Y

r(ii)G

old

(iii)Lo

anstono

nfinancialcorpo

rateclients

SovereignsC

entral

Bank

sandPS

Eswith

amaturity

lt1yr

50


ndequityno

tincludedabove

0

Unencum

beredresid

entia

lmortgages

ofanymaturity

andother


ding

loanstofin

ancialinstitu

tions

with

aremaining

maturity

of1Y

rorgt

1Yrthatw

ould

qualify

forthe

35or

lower

riskweightu

nder

baselIIstand

ardizedapproach

forc

redit

risk

65

Other


allbusinessesh

avingam

aturity

lt1Y

r85

Allothera

ssets

100

Offbalances

heetexpo

sures

Und

rawnam

ount

ofcommitted

creditandliq

uidityfacilities

5

Other

contingent

fund

ingob

ligations

National

superviso

rydiscretio

n






5



50











119889119909119905= 119860119905119909119905119889119905 +

119899

sum

119895=0

119861119895119909119905119892119895(119905 119909119905) 119889119905 + 119886

119905119889119905

+

3

sum

119895=1

119872119895119895(119892 (119905 119909

119905)) 119909119905119889119882119895119895

119905 119909 (119905

0) = 1199090

(23)



119869 (119892) = E [int

1199051

1199050

exp (minus119903119891(119897 minus 1199050)) 119887 (119897 119909

119897 119892 (119897 119909

119897)) 119889119897

+ exp (minus119903119891(1199051minus 1199050)) 1198871(119909 (1199051)) ]

(24)


where 119892 isin G119860 119879 = [119905

0 1199051] and 119887



119887 (119905 119909 119906) = 1198872(1199062) + 1198873(1199091

1199092) (25)

for 1198872 U2rarr R+ and 119887





= 0







119869 (119892) (26)



119869 (119892) (27)



119892isinG119860

119869 (119892) isin G119860 (28)





119869 (119892)


119869 (119892)

(29)

119869 (119892) = E [int

1199051

1199050

exp (minus119903119891(119897 minus 1199050)) [1198872(1199062

119905) + 1198873(119909119905)] d119905

+ exp (minus119903119891(1199051minus 1199050)) 1198871(119909 (1199051)) ]

(30)


3 R+ rarr R+





1198872(1199062) isin 1198622(R)


11986311990621198872(1199062) = minusinfin


11986311990621198872(1199062) = +infin

119863119906211990621198872(1199062) gt 0 forall119906

2isin R

(31)


2



0 = 119863119905V (119905 119909) +

1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909119905)2

]119863119909119909V (119905 119909)

+ 119909 (119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)119863119909V (119905 119909)

+ [1199061

119905minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ 1199062

119905

lowast

119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050)) 1198872(1199062

119905

lowast

)


[119863119909V (119905 119909)]

2

2119863119909119909V (119905 119909)

119903119910

119905

119879

minus1

119905119910

119905

(32)

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050)) 1198871(119909) (33)


0 = 119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))11986311990621198872(1199062

119905) (34)


1198922lowast

(119905 119909) = 1199062lowast

1198922lowast


with 1199062lowast


lowast= minus

119863119909V (119905 119909)

119909119863119909119909V (119905 119909)

minus1

119905119910

119905

1198923119896lowast



(36)



119869lowast= 119869 (119892

lowast) = E [V (119905 119909

0)] (37)



12(119879timesX)

is given by

0 = 119863119905119908 (119905 119909)


[1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2

+(119909)2119879119905]119863119909119909119908 (119905 119909)

+ [ (119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

+119903119910

119905

119879

) 119909]119863119909119908 (119905 119909)

+ [1199061

119905+ 1199062minus 119903119890

119905minus (120590119890)2

]119863119909119908 (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) [1198872(1199062) + 1198873(119909)] ]

119908 (1199051 119909) = exp (minus119903

119891(1199051minus 1199050)) 1198871(119909)

(38)



(b) depending on




inf1199062isinR

1198672(119905 119909 119906

2)

1198672(119905 119909 119906) = 119906

2119863119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) 1198872(1199062)

11986311990621198672(119905 119909 119906) = 119863

119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050))11986311990621198872(1199062) = 0

(39)


2lowast

isin U = R

and

inf1199062isinR

1198672(119905 119909 119906

2) = 119867

2(119905 119909 119906

2lowast

)

= 1199062lowast

119863119909V (119905 119909)


)

(40)




infisinR119896

1198673(119905 119909 )

1198673(119905 119909 ) =

1

2[119879

119905119905119905] (119909)2119863119909119909V (119905 119909)

+119903119910

119905

119879

119909119863119909V (119905 119909)

=1

2(119905+ (

119909 [119863119909V (119905 119909)]

(119909)2119863119909119909V (119905 119909)

) minus1

119905119910

119905)

119879

times 119905(119909)2119863119909119909V (119905 119909)

times (119905+ (

119909 [119863119909V (119905 119909)]

(119909)2119863119909119909V (119905 119909)

) minus1

119905119910

119905)

minus1

2(

[119909119863119909V (119905 119909)]

2

(119909)2119863119909119909V (119905 119909)

)119903119910

119905

119879

minus1

119905119910

119905

119905(119909)2119863119909119909V (119905 119909) gt 0 forall119909 isin R 0

(41)

Hence we have that

lowast= minus

119863119909V (119905 119909)

119909119863119909119909V (119905 119909)

minus1

119905119910

119905

1198923119896lowast

(119905 119909) =

119896lowast if

119896

sum

119894=1

119894lowast

isin [0 1]

119896lowast

sum119896

119895=1119895lowast

if119896

sum

119894=1

119894lowast

gt 1


1198673(119905 119909

lowast) = minus

[119863119909V (119905 119909)]

2

2119863119909119909V (119905 119909)

(119910

119905)119879

minus1

119905119910

119905

(42)





0 = 119863119905V (119905 119909)


1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2

+(119909)2(119879119905) ]119863

119909119909V (119905 119909)

+ [119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+(120590119894)2

+119903119910

119905

119879

] 119909

+ [1199061

119905+ 1199062minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) [1198872(1199062) + 1198873(119909)]

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050)) 1198871(119909)

(43)



lowast=

119869(119892lowast) = E[V(119905

0 1199090)]



119869 (119892) = E [int

1199051

1199050

exp (minus119903119891(119897 minus 1199050))

times [1

21198882((1199062)2

(119897)) +1

21198883(119909119905minus 119897119903)2

] 119889119897

+1

21198881(119909 (1199051) minus 119897119903)2 exp (minus119903

119891(1199051minus 1199050)) ]

(44)


1198871(119909) =

1

21198881(119909 minus 119897

119903)2


1198872(1199062) =

1

21198882(1199062)2


(45)

1198873(119909) =

1

21198883(119909 minus 119897

119903)2

1198883isin (0infin) (46)


minus119905= minus

(119902119905)2

1198882

+ 1198883+ 1199021199052 (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+ 119902119905[minus119903119891minus119903119910

119905

119879

minus1

119905119910

119905+ (120590119890)2

+ (120590119894)2

] 1199021199051= 1198881

(47)

minus 119903

119905= minus

1198883(119909119903

119905minus 119897119903)

119902119905

minus 119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

minus [1199061

119905minus 119903119890

119905minus (120590119890)2

] minus (119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

)

minus (120590119894)2

119909119903(1199051) = 119897119903

(48)

minus 119897119905= minus119903119891119897119905+ 1198883(119909119903

119905minus 119897119903)2

minus 119902119905(120590119890)2

(119909119903

119905minus 1)2

minus 119902119905(120590119894)2

(119909119903

119905)2

119897 (1199051) = 0

(49)





1199062lowast

119905= minus

(119909 minus 119909119903

119905) 119902119905

1198882

1198922lowast

(119905 119909) = 1199062lowast

1198922lowast


lowast

119905= minus

(119909 minus 119909119903

119905)

119909minus1

119905119910

119905 1198923lowast

119879 timesX 997888rarr R119896

(50)

1198923119896lowast




(51)


V (119905 119909) = exp (minus119903119891(1199051minus 1199050)) [

1

2(119909 minus 119909

119903

119905)2

119902119905+

1

2119897119905] (52)

119869lowast= 119869 (119892

lowast) = E [V (119905

0 1199090)] (53)




minus119905= minus119887(119902

119905)2

+ 119888 + 2119886119905119902119905 1199021199051= 1198881

119887 =1

1198882 119888 = 119888

3isin (0infin) 119886 119879 997888rarr R

(54)


1199021

1199051= 119902 (119905

1minus 119905) 119886

1

119905= 119886 (119905

1minus 119905) 119902

1 [0 1199051minus 1199050] 997888rarr R

1

119905= minus (119905

1minus 119905) = minus119887(119902

1

119905)2

+ 119888 + 21198861

1199051199021

119905 1199021

1199050= 1198881

0 = 1

119905+ 119887(1199021

119905)2

minus 119888 minus 21198861

119905119902119905 1199021

0= 1198881

(55)


minus2

119905= minus119887(119902

2

119905)2

+ 119888 1199022

1199051= 1198881 119887 119888 isin (0infin) (56)


119905gt 0 for all 119905 isin 119879 = [0 119905

1minus 1199050]


1minus 1199050] 1199021119905

ge

1199022

119905gt 0 if


0 119887) ge (


119905

minus1198861

119905119887


(57)



(119886119905)2


1

1198882minus

1

1198882)

(58)


=


1minus 1199050]




1

1198882minus

1

1198882ge 0

(119886119905)2

le 1198884le (119888 minus 119888) (119887 minus 119887) = (119888

3minus 1198883) (

1

1198882minus

1

1198882)

(59)


1minus 1199050] 119902(1199051minus

119905) = 1199021

119905ge 1199022

119905gt 0




Note first that

V (119905 119909) = exp (minus119903119891(119905 minus 1199050)) [

1

2(119909 minus 119909

119903

119905)2

119902119905+

1

2119897119905]

119863119905V (119905 119909) = minus119903

119891V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))


119905) 119903

119905119902119905+

1

2(119909 minus 119909

119903

119905)2

119905+

1

2

119897119905]

119863119909V (119905 119909) = exp (minus119903

119891(119905 minus 1199050)) (119909 minus 119909

119903

119905) 119902119905

119863119909119909V (119905 119909) = exp (minus119903

119891(119905 minus 1199050)) 119902119905

(60)


So

0 = 119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))11986311990621198872(1199062)


119903

119905) 119902119905+ 11988821199062]

1199062lowast

= minus(119909 minus 119909

119903

119905) 119902119905

1198882

lowast= minus

(119909 minus 119909119903

119905) 119902119905

119909119902119905

minus1

119905119910

119905

(61)


1199021199051= 1198881 119909

119903(1199051) = 119897119903 119897 (119905

1) = 0

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050))

times [1

2(119909 minus 119909

119903(1199051))2

1199021199051+

1

2119897 (1199051)]

= exp (minus119903119891(1199051minus 1199050))

1

21198881(119909 minus 119897

119903)2

= exp (minus119903119891(1199051minus 1199050)) 1198871(119909)


exp (+119903119891(119905 minus 1199050))

times [119863119905V (119905 119909) +

1

2[(120590119890)2

(1 minus 119909)2

+(120590119894)2

(119909)2]119863119909119909V (119905 119909)

+ 119909 [119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]119863119909V (119905 119909)

+ [1199061

119905minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ 1199062lowast

119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050)) 1198872(1199062lowast

)

+ exp (minus119903119891(119905 minus 1199050)) 1198873(119909)

minus1

2

(119863119909V (119905 119909))

2

119863119909119909V (119905 119909)

119903119910

119905

119879

minus1

119905119910

119905]

= minus119903119891 1

2(119909 minus 119909

119903

119905)2

119902119905minus

1

2119903119891119897119905minus (119909 minus 119909

119903

119905) 119902119905119903

119905

+1

2(119909 minus 119909

119903

119905)2

119905+

1

2

119897119905

+1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2] 119902119905

+ (119909 minus 119909119903

119905) 119902119905[119909 (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+1199061

119905minus 119903119890

119905minus (120590119890)2

]

minus1

2

(119909 minus 119909119903

119905)2

(119902119905)2

1198882

+1

21198883(119909 minus 119897

119903)2

minus1

2(119909 minus 119909

119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

=1

2(119909)2[ minus 119903119891119902119905+ 119905+ (120590119890)2

119902119905+ (120590119894)2

119902119905

+ 2119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

minus(119902119905)2

1198882

+ 1198883minus 119902119905

119903119910

119905

119879

minus1

119905119910

119905]

+ 119909 [ + 119903119891119909119903

119905119902119905minus 119902119905119903

119905minus 119909119903

119905119905minus (120590119890)2

119902119905

minus 119909119903

119905119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ 119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] +119909119903

119905(119902119905)2

1198882

minus1198883119897119903+ 119909119903

119905119902119905

119903119910

119905

119879

minus1

119905119910

119905]

+1

2(119909)0[ minus 119903119891(119909119903

119905)2

119902119905+ 2119909119903

119905119902119905119903

119905+ (119909119903

119905)2

119905minus 119903119891119897119905

+ (120590119890)2

119902119905minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

]

minus(119909119903

119905)2

(119902119905)2

1198882

+ 1198883(119897119903)2

minus(119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905+ 119897119905]

(62)



minus 119902119905119903

119905minus 119909119903

119905119905+ 119903119891119909119903

119905119902119905minus 2(120590119890)2

119902119905

minus 119909119903

119905119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ 119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

]

+119909119903

119905(119902119905)2

1198882

minus 1198883119897119903+ 119909119903

119905119902119905

119903119910

119905

119879

minus1

119905119910

119905

= minus119902119905119903

119905

+ 119909119903

119905[minus

(119902119905)2

1198882

+ 1198883

+ 2119902119905(119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+119902119905((120590119890)2

+ (120590119894)2

minus 119903119891minus119903119910

119905

119879

minus1

119905119910

119905)]

minus 119909119903

119905119902119905[ (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

minus119903119891minus119903119910

119905

119879

minus1

119905119910

119905]

+ 119902 [minus(120590119890)2

+ (1199061

119905minus 119903119890

119905minus (120590119890)2

)] +119909119903

119905(119902119905)2

1198882

minus 1198883119897119903

= 119902119905[minus119903

119905+

1198883(119909119903

119905minus 119897119903)

119902119905

]

+ 119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ (119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

) + (120590119894)2

+ (1199061

119905minus 119903119890

119905minus (120590119890)2

)

= 0

(63)



119897119905minus 119903119891119897119905minus 119903119891(119909119903

119905)2

119902119905

+ 2119909119903

119905[119902119905119903

119905+ 119909119903

119905119905] minus (119909

119903

119905)2

119905+ (120590119890)2

119902119905

minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] + 1198883(119897119903)2

minus(119909119903

119905)2

(119902119905)2

1198882

minus (119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905


119897119905minus 119903119891119897119905minus 119903119891(119909119903

119905)2

119902119905+ (120590119890)2

119902119905

minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] + 1198883(119897119903)2

minus(119909119903

119905)2

(119902119905)2

1198882

minus (119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

+ 2119903119891(119909119903

119905)2

119902119905minus 2119909119903

119905119902119905(120590119890)2

minus 2(119909119903

119905)2

119902119905[119903

T+ 119903119890minus 119903119894+ (120590119890)2

+ (120590119894)2

]

+ 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] minus 11988831198971199032119909119903

119905

+2(119909119903

119905)2

(119902119905)2

1198882

+ 2(119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

minus(119909119903

119905)2

(119902119905)2

1198882

+ 1198883(119909119903

119905)2

+ (119909119903

119905)2

1199021199052 (119903

T+ 119903119890minus 119903119894+ (120590119890)2

+ (120590119894)2

)

+ (119909119903

119905)2

119902119905(minus119903119891+ (120590119890)2

+ (120590119894)2

minus119903119910

119905

119879

minus1

119905119910

119905)

= 119897119905minus 119903119891119897119905+ 1198883(119909119903

119905minus 119897119903)2

minus (120590119890)2

119902119905(119909119903

119905minus 1)2

minus (120590119894)2

119902119905(119909119903

119905)2

= 0

(64)







2gt 0 should


2(1199062) = (12)119888

2(1199062)2 given by (45) This







3(119909) = (12)119888



1198873(119909) = 119888

3[exp (119909 minus 119897

119903) + (119897119903minus 119909) minus 1] (65)







1199051198882 which





119903(119909) = (12)119888

3(119909 minus 119909




119903



119905119910



minus1

119905119910


deviation (119909 minus 119909119903

119905)




1199061

119905minus 119903119890

119905minus (120590119890)2

(66)



119890

119905is the


119903119890



1199051 119909119905lt 119897119903The term 119888

3(119909119903





119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

] (67)




T(119905) + (120590

119890)2+ (120590119894)2 is the effect of


(119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

) minus (120590119894)2

(68)




minus119903

119905= minus119896 (119909

119903

119905minus 119898) 119909

119903(1199051) = 119897119903

119896 = (119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2

+ (120590119894)2

)) +1198883

1199020

119898 =


) + (120590119890)2

(119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2+ (120590119894)2

)) + 11988831199020

(69)


119902119905

= 1199020 and lim

119905darr0119909119903




1199062lowast

119905= minus

(119909119905minus 119898) 119902

0

1198882

= gt 0 if 119909

119905lt 119898

lt 0 if 119909119905gt 119898

120587lowast

119905= minus

(119909119905minus 119898)

119909119905

119895

= gt 0 if 119909

119905lt 119898


(70)


Case 1 (119909119905











119898 =


) + (120590119890)2

(119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2+ (120590119894)2

)) + 11988831199020

(71)





Appendices













References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





Time (years)


0 01 02 03 04 05 06 07 08 09 1

5

0

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

minus45



Parameter Value119862 500119903119894 001119910 0061199062 003

119903119862 005120590119890 16

04 150119903119890 004120590119894 18

1199061 001

119882 003











119905in (4)















(22)


Table4Summaryof

netstablefun

ding

ratio

Availables

tablefun

ding

(sou

rces)

Requ

iredstablefund

ing(uses)

Item

Avlblty

Fctr

Item

Rqrd

Fctr

(i)T1KandT2

Kinstr

uments

(ii)O

ther

preferredshares

andcapitalinstrum

entsin

excessof

T2Kallowableam

ount

having

aneffectiv

ematurity

of1Y

rorgt

1Yr

(iii)Other

liabilitiesw

ithan

effectiv

ematurity

of1Y

rorgt

1Yr

100

(i)Ca

sh(ii)S

hort-te

rmun

securedactiv

elytraded

instr

uments(lt1Y

r)(iii)Securitiesw

ithexactly

offsetting

reverser

epo

(iv)S

ecurities

with

remaining

maturity

lt1Y

r(v)N


ancialsw

ithremaining

maturity

lt1Y

r

0

Stabledepo

sitso

fretailand

smallbusinessc

ustomers

(non

maturity

orresid

ualm

aturity

lt1Y

r)90

Debtissuedor

guaranteed


sBISIM

FEC

non

centralgovernm


entb

anks

with

a0ris

kweightu

nder

BaselIIS

tand

ardizedAp

proach

5

Lessstabledepo

sitso

fretailand

smallbusinessc

ustomers

(non

maturity

orresid

ualm

aturity

lt1Y

r)80

Unencum

beredno

nfinancialsenioru

nsecured

corporateb

onds

and

coveredbo

ndsrated

atleastA

Aminusanddebt

thatisissuedby

SovereignsC

entralBa

nksandPS

Eswith

arisk

weightin

gof

20

maturity

ge1Y

r

20

Who

lesalefund

ingprovided

byno

nfinancialcorpo

ratecusto

mers


ultilateraldevelopm

entb

anksand

PSEs

(non

maturity

orresid

ualm

aturity

lt1Y

r)50

(i)Unencum


rnon

financialsenior

unsecuredcorporateb

onds

(orc

overed

bond

s)ratedfro

mA+to

Aminus

maturity

ge1Y

r(ii)G

old

(iii)Lo

anstono

nfinancialcorpo

rateclients

SovereignsC

entral

Bank

sandPS

Eswith

amaturity

lt1yr

50


ndequityno

tincludedabove

0

Unencum

beredresid

entia

lmortgages

ofanymaturity

andother


ding

loanstofin

ancialinstitu

tions

with

aremaining

maturity

of1Y

rorgt

1Yrthatw

ould

qualify

forthe

35or

lower

riskweightu

nder

baselIIstand

ardizedapproach

forc

redit

risk

65

Other


allbusinessesh

avingam

aturity

lt1Y

r85

Allothera

ssets

100

Offbalances

heetexpo

sures

Und

rawnam

ount

ofcommitted

creditandliq

uidityfacilities

5

Other

contingent

fund

ingob

ligations

National

superviso

rydiscretio

n






5



50











119889119909119905= 119860119905119909119905119889119905 +

119899

sum

119895=0

119861119895119909119905119892119895(119905 119909119905) 119889119905 + 119886

119905119889119905

+

3

sum

119895=1

119872119895119895(119892 (119905 119909

119905)) 119909119905119889119882119895119895

119905 119909 (119905

0) = 1199090

(23)



119869 (119892) = E [int

1199051

1199050

exp (minus119903119891(119897 minus 1199050)) 119887 (119897 119909

119897 119892 (119897 119909

119897)) 119889119897

+ exp (minus119903119891(1199051minus 1199050)) 1198871(119909 (1199051)) ]

(24)


where 119892 isin G119860 119879 = [119905

0 1199051] and 119887



119887 (119905 119909 119906) = 1198872(1199062) + 1198873(1199091

1199092) (25)

for 1198872 U2rarr R+ and 119887





= 0







119869 (119892) (26)



119869 (119892) (27)



119892isinG119860

119869 (119892) isin G119860 (28)





119869 (119892)


119869 (119892)

(29)

119869 (119892) = E [int

1199051

1199050

exp (minus119903119891(119897 minus 1199050)) [1198872(1199062

119905) + 1198873(119909119905)] d119905

+ exp (minus119903119891(1199051minus 1199050)) 1198871(119909 (1199051)) ]

(30)


3 R+ rarr R+





1198872(1199062) isin 1198622(R)


11986311990621198872(1199062) = minusinfin


11986311990621198872(1199062) = +infin

119863119906211990621198872(1199062) gt 0 forall119906

2isin R

(31)


2



0 = 119863119905V (119905 119909) +

1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909119905)2

]119863119909119909V (119905 119909)

+ 119909 (119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)119863119909V (119905 119909)

+ [1199061

119905minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ 1199062

119905

lowast

119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050)) 1198872(1199062

119905

lowast

)


[119863119909V (119905 119909)]

2

2119863119909119909V (119905 119909)

119903119910

119905

119879

minus1

119905119910

119905

(32)

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050)) 1198871(119909) (33)


0 = 119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))11986311990621198872(1199062

119905) (34)


1198922lowast

(119905 119909) = 1199062lowast

1198922lowast


with 1199062lowast


lowast= minus

119863119909V (119905 119909)

119909119863119909119909V (119905 119909)

minus1

119905119910

119905

1198923119896lowast



(36)



119869lowast= 119869 (119892

lowast) = E [V (119905 119909

0)] (37)



12(119879timesX)

is given by

0 = 119863119905119908 (119905 119909)


[1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2

+(119909)2119879119905]119863119909119909119908 (119905 119909)

+ [ (119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

+119903119910

119905

119879

) 119909]119863119909119908 (119905 119909)

+ [1199061

119905+ 1199062minus 119903119890

119905minus (120590119890)2

]119863119909119908 (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) [1198872(1199062) + 1198873(119909)] ]

119908 (1199051 119909) = exp (minus119903

119891(1199051minus 1199050)) 1198871(119909)

(38)



(b) depending on




inf1199062isinR

1198672(119905 119909 119906

2)

1198672(119905 119909 119906) = 119906

2119863119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) 1198872(1199062)

11986311990621198672(119905 119909 119906) = 119863

119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050))11986311990621198872(1199062) = 0

(39)


2lowast

isin U = R

and

inf1199062isinR

1198672(119905 119909 119906

2) = 119867

2(119905 119909 119906

2lowast

)

= 1199062lowast

119863119909V (119905 119909)


)

(40)




infisinR119896

1198673(119905 119909 )

1198673(119905 119909 ) =

1

2[119879

119905119905119905] (119909)2119863119909119909V (119905 119909)

+119903119910

119905

119879

119909119863119909V (119905 119909)

=1

2(119905+ (

119909 [119863119909V (119905 119909)]

(119909)2119863119909119909V (119905 119909)

) minus1

119905119910

119905)

119879

times 119905(119909)2119863119909119909V (119905 119909)

times (119905+ (

119909 [119863119909V (119905 119909)]

(119909)2119863119909119909V (119905 119909)

) minus1

119905119910

119905)

minus1

2(

[119909119863119909V (119905 119909)]

2

(119909)2119863119909119909V (119905 119909)

)119903119910

119905

119879

minus1

119905119910

119905

119905(119909)2119863119909119909V (119905 119909) gt 0 forall119909 isin R 0

(41)

Hence we have that

lowast= minus

119863119909V (119905 119909)

119909119863119909119909V (119905 119909)

minus1

119905119910

119905

1198923119896lowast

(119905 119909) =

119896lowast if

119896

sum

119894=1

119894lowast

isin [0 1]

119896lowast

sum119896

119895=1119895lowast

if119896

sum

119894=1

119894lowast

gt 1


1198673(119905 119909

lowast) = minus

[119863119909V (119905 119909)]

2

2119863119909119909V (119905 119909)

(119910

119905)119879

minus1

119905119910

119905

(42)





0 = 119863119905V (119905 119909)


1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2

+(119909)2(119879119905) ]119863

119909119909V (119905 119909)

+ [119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+(120590119894)2

+119903119910

119905

119879

] 119909

+ [1199061

119905+ 1199062minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) [1198872(1199062) + 1198873(119909)]

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050)) 1198871(119909)

(43)



lowast=

119869(119892lowast) = E[V(119905

0 1199090)]



119869 (119892) = E [int

1199051

1199050

exp (minus119903119891(119897 minus 1199050))

times [1

21198882((1199062)2

(119897)) +1

21198883(119909119905minus 119897119903)2

] 119889119897

+1

21198881(119909 (1199051) minus 119897119903)2 exp (minus119903

119891(1199051minus 1199050)) ]

(44)


1198871(119909) =

1

21198881(119909 minus 119897

119903)2


1198872(1199062) =

1

21198882(1199062)2


(45)

1198873(119909) =

1

21198883(119909 minus 119897

119903)2

1198883isin (0infin) (46)


minus119905= minus

(119902119905)2

1198882

+ 1198883+ 1199021199052 (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+ 119902119905[minus119903119891minus119903119910

119905

119879

minus1

119905119910

119905+ (120590119890)2

+ (120590119894)2

] 1199021199051= 1198881

(47)

minus 119903

119905= minus

1198883(119909119903

119905minus 119897119903)

119902119905

minus 119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

minus [1199061

119905minus 119903119890

119905minus (120590119890)2

] minus (119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

)

minus (120590119894)2

119909119903(1199051) = 119897119903

(48)

minus 119897119905= minus119903119891119897119905+ 1198883(119909119903

119905minus 119897119903)2

minus 119902119905(120590119890)2

(119909119903

119905minus 1)2

minus 119902119905(120590119894)2

(119909119903

119905)2

119897 (1199051) = 0

(49)





1199062lowast

119905= minus

(119909 minus 119909119903

119905) 119902119905

1198882

1198922lowast

(119905 119909) = 1199062lowast

1198922lowast


lowast

119905= minus

(119909 minus 119909119903

119905)

119909minus1

119905119910

119905 1198923lowast

119879 timesX 997888rarr R119896

(50)

1198923119896lowast




(51)


V (119905 119909) = exp (minus119903119891(1199051minus 1199050)) [

1

2(119909 minus 119909

119903

119905)2

119902119905+

1

2119897119905] (52)

119869lowast= 119869 (119892

lowast) = E [V (119905

0 1199090)] (53)




minus119905= minus119887(119902

119905)2

+ 119888 + 2119886119905119902119905 1199021199051= 1198881

119887 =1

1198882 119888 = 119888

3isin (0infin) 119886 119879 997888rarr R

(54)


1199021

1199051= 119902 (119905

1minus 119905) 119886

1

119905= 119886 (119905

1minus 119905) 119902

1 [0 1199051minus 1199050] 997888rarr R

1

119905= minus (119905

1minus 119905) = minus119887(119902

1

119905)2

+ 119888 + 21198861

1199051199021

119905 1199021

1199050= 1198881

0 = 1

119905+ 119887(1199021

119905)2

minus 119888 minus 21198861

119905119902119905 1199021

0= 1198881

(55)


minus2

119905= minus119887(119902

2

119905)2

+ 119888 1199022

1199051= 1198881 119887 119888 isin (0infin) (56)


119905gt 0 for all 119905 isin 119879 = [0 119905

1minus 1199050]


1minus 1199050] 1199021119905

ge

1199022

119905gt 0 if


0 119887) ge (


119905

minus1198861

119905119887


(57)



(119886119905)2


1

1198882minus

1

1198882)

(58)


=


1minus 1199050]




1

1198882minus

1

1198882ge 0

(119886119905)2

le 1198884le (119888 minus 119888) (119887 minus 119887) = (119888

3minus 1198883) (

1

1198882minus

1

1198882)

(59)


1minus 1199050] 119902(1199051minus

119905) = 1199021

119905ge 1199022

119905gt 0




Note first that

V (119905 119909) = exp (minus119903119891(119905 minus 1199050)) [

1

2(119909 minus 119909

119903

119905)2

119902119905+

1

2119897119905]

119863119905V (119905 119909) = minus119903

119891V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))


119905) 119903

119905119902119905+

1

2(119909 minus 119909

119903

119905)2

119905+

1

2

119897119905]

119863119909V (119905 119909) = exp (minus119903

119891(119905 minus 1199050)) (119909 minus 119909

119903

119905) 119902119905

119863119909119909V (119905 119909) = exp (minus119903

119891(119905 minus 1199050)) 119902119905

(60)


So

0 = 119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))11986311990621198872(1199062)


119903

119905) 119902119905+ 11988821199062]

1199062lowast

= minus(119909 minus 119909

119903

119905) 119902119905

1198882

lowast= minus

(119909 minus 119909119903

119905) 119902119905

119909119902119905

minus1

119905119910

119905

(61)


1199021199051= 1198881 119909

119903(1199051) = 119897119903 119897 (119905

1) = 0

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050))

times [1

2(119909 minus 119909

119903(1199051))2

1199021199051+

1

2119897 (1199051)]

= exp (minus119903119891(1199051minus 1199050))

1

21198881(119909 minus 119897

119903)2

= exp (minus119903119891(1199051minus 1199050)) 1198871(119909)


exp (+119903119891(119905 minus 1199050))

times [119863119905V (119905 119909) +

1

2[(120590119890)2

(1 minus 119909)2

+(120590119894)2

(119909)2]119863119909119909V (119905 119909)

+ 119909 [119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]119863119909V (119905 119909)

+ [1199061

119905minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ 1199062lowast

119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050)) 1198872(1199062lowast

)

+ exp (minus119903119891(119905 minus 1199050)) 1198873(119909)

minus1

2

(119863119909V (119905 119909))

2

119863119909119909V (119905 119909)

119903119910

119905

119879

minus1

119905119910

119905]

= minus119903119891 1

2(119909 minus 119909

119903

119905)2

119902119905minus

1

2119903119891119897119905minus (119909 minus 119909

119903

119905) 119902119905119903

119905

+1

2(119909 minus 119909

119903

119905)2

119905+

1

2

119897119905

+1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2] 119902119905

+ (119909 minus 119909119903

119905) 119902119905[119909 (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+1199061

119905minus 119903119890

119905minus (120590119890)2

]

minus1

2

(119909 minus 119909119903

119905)2

(119902119905)2

1198882

+1

21198883(119909 minus 119897

119903)2

minus1

2(119909 minus 119909

119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

=1

2(119909)2[ minus 119903119891119902119905+ 119905+ (120590119890)2

119902119905+ (120590119894)2

119902119905

+ 2119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

minus(119902119905)2

1198882

+ 1198883minus 119902119905

119903119910

119905

119879

minus1

119905119910

119905]

+ 119909 [ + 119903119891119909119903

119905119902119905minus 119902119905119903

119905minus 119909119903

119905119905minus (120590119890)2

119902119905

minus 119909119903

119905119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ 119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] +119909119903

119905(119902119905)2

1198882

minus1198883119897119903+ 119909119903

119905119902119905

119903119910

119905

119879

minus1

119905119910

119905]

+1

2(119909)0[ minus 119903119891(119909119903

119905)2

119902119905+ 2119909119903

119905119902119905119903

119905+ (119909119903

119905)2

119905minus 119903119891119897119905

+ (120590119890)2

119902119905minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

]

minus(119909119903

119905)2

(119902119905)2

1198882

+ 1198883(119897119903)2

minus(119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905+ 119897119905]

(62)



minus 119902119905119903

119905minus 119909119903

119905119905+ 119903119891119909119903

119905119902119905minus 2(120590119890)2

119902119905

minus 119909119903

119905119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ 119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

]

+119909119903

119905(119902119905)2

1198882

minus 1198883119897119903+ 119909119903

119905119902119905

119903119910

119905

119879

minus1

119905119910

119905

= minus119902119905119903

119905

+ 119909119903

119905[minus

(119902119905)2

1198882

+ 1198883

+ 2119902119905(119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+119902119905((120590119890)2

+ (120590119894)2

minus 119903119891minus119903119910

119905

119879

minus1

119905119910

119905)]

minus 119909119903

119905119902119905[ (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

minus119903119891minus119903119910

119905

119879

minus1

119905119910

119905]

+ 119902 [minus(120590119890)2

+ (1199061

119905minus 119903119890

119905minus (120590119890)2

)] +119909119903

119905(119902119905)2

1198882

minus 1198883119897119903

= 119902119905[minus119903

119905+

1198883(119909119903

119905minus 119897119903)

119902119905

]

+ 119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ (119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

) + (120590119894)2

+ (1199061

119905minus 119903119890

119905minus (120590119890)2

)

= 0

(63)



119897119905minus 119903119891119897119905minus 119903119891(119909119903

119905)2

119902119905

+ 2119909119903

119905[119902119905119903

119905+ 119909119903

119905119905] minus (119909

119903

119905)2

119905+ (120590119890)2

119902119905

minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] + 1198883(119897119903)2

minus(119909119903

119905)2

(119902119905)2

1198882

minus (119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905


119897119905minus 119903119891119897119905minus 119903119891(119909119903

119905)2

119902119905+ (120590119890)2

119902119905

minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] + 1198883(119897119903)2

minus(119909119903

119905)2

(119902119905)2

1198882

minus (119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

+ 2119903119891(119909119903

119905)2

119902119905minus 2119909119903

119905119902119905(120590119890)2

minus 2(119909119903

119905)2

119902119905[119903

T+ 119903119890minus 119903119894+ (120590119890)2

+ (120590119894)2

]

+ 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] minus 11988831198971199032119909119903

119905

+2(119909119903

119905)2

(119902119905)2

1198882

+ 2(119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

minus(119909119903

119905)2

(119902119905)2

1198882

+ 1198883(119909119903

119905)2

+ (119909119903

119905)2

1199021199052 (119903

T+ 119903119890minus 119903119894+ (120590119890)2

+ (120590119894)2

)

+ (119909119903

119905)2

119902119905(minus119903119891+ (120590119890)2

+ (120590119894)2

minus119903119910

119905

119879

minus1

119905119910

119905)

= 119897119905minus 119903119891119897119905+ 1198883(119909119903

119905minus 119897119903)2

minus (120590119890)2

119902119905(119909119903

119905minus 1)2

minus (120590119894)2

119902119905(119909119903

119905)2

= 0

(64)







2gt 0 should


2(1199062) = (12)119888

2(1199062)2 given by (45) This







3(119909) = (12)119888



1198873(119909) = 119888

3[exp (119909 minus 119897

119903) + (119897119903minus 119909) minus 1] (65)







1199051198882 which





119903(119909) = (12)119888

3(119909 minus 119909




119903



119905119910



minus1

119905119910


deviation (119909 minus 119909119903

119905)




1199061

119905minus 119903119890

119905minus (120590119890)2

(66)



119890

119905is the


119903119890



1199051 119909119905lt 119897119903The term 119888

3(119909119903





119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

] (67)




T(119905) + (120590

119890)2+ (120590119894)2 is the effect of


(119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

) minus (120590119894)2

(68)




minus119903

119905= minus119896 (119909

119903

119905minus 119898) 119909

119903(1199051) = 119897119903

119896 = (119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2

+ (120590119894)2

)) +1198883

1199020

119898 =


) + (120590119890)2

(119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2+ (120590119894)2

)) + 11988831199020

(69)


119902119905

= 1199020 and lim

119905darr0119909119903




1199062lowast

119905= minus

(119909119905minus 119898) 119902

0

1198882

= gt 0 if 119909

119905lt 119898

lt 0 if 119909119905gt 119898

120587lowast

119905= minus

(119909119905minus 119898)

119909119905

119895

= gt 0 if 119909

119905lt 119898


(70)


Case 1 (119909119905











119898 =


) + (120590119890)2

(119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2+ (120590119894)2

)) + 11988831199020

(71)





Appendices













References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




Table4Summaryof

netstablefun

ding

ratio

Availables

tablefun

ding

(sou

rces)

Requ

iredstablefund

ing(uses)

Item

Avlblty

Fctr

Item

Rqrd

Fctr

(i)T1KandT2

Kinstr

uments

(ii)O

ther

preferredshares

andcapitalinstrum

entsin

excessof

T2Kallowableam

ount

having

aneffectiv

ematurity

of1Y

rorgt

1Yr

(iii)Other

liabilitiesw

ithan

effectiv

ematurity

of1Y

rorgt

1Yr

100

(i)Ca

sh(ii)S

hort-te

rmun

securedactiv

elytraded

instr

uments(lt1Y

r)(iii)Securitiesw

ithexactly

offsetting

reverser

epo

(iv)S

ecurities

with

remaining

maturity

lt1Y

r(v)N


ancialsw

ithremaining

maturity

lt1Y

r

0

Stabledepo

sitso

fretailand

smallbusinessc

ustomers

(non

maturity

orresid

ualm

aturity

lt1Y

r)90

Debtissuedor

guaranteed


sBISIM

FEC

non

centralgovernm


entb

anks

with

a0ris

kweightu

nder

BaselIIS

tand

ardizedAp

proach

5

Lessstabledepo

sitso

fretailand

smallbusinessc

ustomers

(non

maturity

orresid

ualm

aturity

lt1Y

r)80

Unencum

beredno

nfinancialsenioru

nsecured

corporateb

onds

and

coveredbo

ndsrated

atleastA

Aminusanddebt

thatisissuedby

SovereignsC

entralBa

nksandPS

Eswith

arisk

weightin

gof

20

maturity

ge1Y

r

20

Who

lesalefund

ingprovided

byno

nfinancialcorpo

ratecusto

mers


ultilateraldevelopm

entb

anksand

PSEs

(non

maturity

orresid

ualm

aturity

lt1Y

r)50

(i)Unencum


rnon

financialsenior

unsecuredcorporateb

onds

(orc

overed

bond

s)ratedfro

mA+to

Aminus

maturity

ge1Y

r(ii)G

old

(iii)Lo

anstono

nfinancialcorpo

rateclients

SovereignsC

entral

Bank

sandPS

Eswith

amaturity

lt1yr

50


ndequityno

tincludedabove

0

Unencum

beredresid

entia

lmortgages

ofanymaturity

andother


ding

loanstofin

ancialinstitu

tions

with

aremaining

maturity

of1Y

rorgt

1Yrthatw

ould

qualify

forthe

35or

lower

riskweightu

nder

baselIIstand

ardizedapproach

forc

redit

risk

65

Other


allbusinessesh

avingam

aturity

lt1Y

r85

Allothera

ssets

100

Offbalances

heetexpo

sures

Und

rawnam

ount

ofcommitted

creditandliq

uidityfacilities

5

Other

contingent

fund

ingob

ligations

National

superviso

rydiscretio

n






5



50











119889119909119905= 119860119905119909119905119889119905 +

119899

sum

119895=0

119861119895119909119905119892119895(119905 119909119905) 119889119905 + 119886

119905119889119905

+

3

sum

119895=1

119872119895119895(119892 (119905 119909

119905)) 119909119905119889119882119895119895

119905 119909 (119905

0) = 1199090

(23)



119869 (119892) = E [int

1199051

1199050

exp (minus119903119891(119897 minus 1199050)) 119887 (119897 119909

119897 119892 (119897 119909

119897)) 119889119897

+ exp (minus119903119891(1199051minus 1199050)) 1198871(119909 (1199051)) ]

(24)


where 119892 isin G119860 119879 = [119905

0 1199051] and 119887



119887 (119905 119909 119906) = 1198872(1199062) + 1198873(1199091

1199092) (25)

for 1198872 U2rarr R+ and 119887





= 0







119869 (119892) (26)



119869 (119892) (27)



119892isinG119860

119869 (119892) isin G119860 (28)





119869 (119892)


119869 (119892)

(29)

119869 (119892) = E [int

1199051

1199050

exp (minus119903119891(119897 minus 1199050)) [1198872(1199062

119905) + 1198873(119909119905)] d119905

+ exp (minus119903119891(1199051minus 1199050)) 1198871(119909 (1199051)) ]

(30)


3 R+ rarr R+





1198872(1199062) isin 1198622(R)


11986311990621198872(1199062) = minusinfin


11986311990621198872(1199062) = +infin

119863119906211990621198872(1199062) gt 0 forall119906

2isin R

(31)


2



0 = 119863119905V (119905 119909) +

1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909119905)2

]119863119909119909V (119905 119909)

+ 119909 (119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)119863119909V (119905 119909)

+ [1199061

119905minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ 1199062

119905

lowast

119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050)) 1198872(1199062

119905

lowast

)


[119863119909V (119905 119909)]

2

2119863119909119909V (119905 119909)

119903119910

119905

119879

minus1

119905119910

119905

(32)

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050)) 1198871(119909) (33)


0 = 119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))11986311990621198872(1199062

119905) (34)


1198922lowast

(119905 119909) = 1199062lowast

1198922lowast


with 1199062lowast


lowast= minus

119863119909V (119905 119909)

119909119863119909119909V (119905 119909)

minus1

119905119910

119905

1198923119896lowast



(36)



119869lowast= 119869 (119892

lowast) = E [V (119905 119909

0)] (37)



12(119879timesX)

is given by

0 = 119863119905119908 (119905 119909)


[1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2

+(119909)2119879119905]119863119909119909119908 (119905 119909)

+ [ (119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

+119903119910

119905

119879

) 119909]119863119909119908 (119905 119909)

+ [1199061

119905+ 1199062minus 119903119890

119905minus (120590119890)2

]119863119909119908 (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) [1198872(1199062) + 1198873(119909)] ]

119908 (1199051 119909) = exp (minus119903

119891(1199051minus 1199050)) 1198871(119909)

(38)



(b) depending on




inf1199062isinR

1198672(119905 119909 119906

2)

1198672(119905 119909 119906) = 119906

2119863119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) 1198872(1199062)

11986311990621198672(119905 119909 119906) = 119863

119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050))11986311990621198872(1199062) = 0

(39)


2lowast

isin U = R

and

inf1199062isinR

1198672(119905 119909 119906

2) = 119867

2(119905 119909 119906

2lowast

)

= 1199062lowast

119863119909V (119905 119909)


)

(40)




infisinR119896

1198673(119905 119909 )

1198673(119905 119909 ) =

1

2[119879

119905119905119905] (119909)2119863119909119909V (119905 119909)

+119903119910

119905

119879

119909119863119909V (119905 119909)

=1

2(119905+ (

119909 [119863119909V (119905 119909)]

(119909)2119863119909119909V (119905 119909)

) minus1

119905119910

119905)

119879

times 119905(119909)2119863119909119909V (119905 119909)

times (119905+ (

119909 [119863119909V (119905 119909)]

(119909)2119863119909119909V (119905 119909)

) minus1

119905119910

119905)

minus1

2(

[119909119863119909V (119905 119909)]

2

(119909)2119863119909119909V (119905 119909)

)119903119910

119905

119879

minus1

119905119910

119905

119905(119909)2119863119909119909V (119905 119909) gt 0 forall119909 isin R 0

(41)

Hence we have that

lowast= minus

119863119909V (119905 119909)

119909119863119909119909V (119905 119909)

minus1

119905119910

119905

1198923119896lowast

(119905 119909) =

119896lowast if

119896

sum

119894=1

119894lowast

isin [0 1]

119896lowast

sum119896

119895=1119895lowast

if119896

sum

119894=1

119894lowast

gt 1


1198673(119905 119909

lowast) = minus

[119863119909V (119905 119909)]

2

2119863119909119909V (119905 119909)

(119910

119905)119879

minus1

119905119910

119905

(42)





0 = 119863119905V (119905 119909)


1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2

+(119909)2(119879119905) ]119863

119909119909V (119905 119909)

+ [119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+(120590119894)2

+119903119910

119905

119879

] 119909

+ [1199061

119905+ 1199062minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) [1198872(1199062) + 1198873(119909)]

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050)) 1198871(119909)

(43)



lowast=

119869(119892lowast) = E[V(119905

0 1199090)]



119869 (119892) = E [int

1199051

1199050

exp (minus119903119891(119897 minus 1199050))

times [1

21198882((1199062)2

(119897)) +1

21198883(119909119905minus 119897119903)2

] 119889119897

+1

21198881(119909 (1199051) minus 119897119903)2 exp (minus119903

119891(1199051minus 1199050)) ]

(44)


1198871(119909) =

1

21198881(119909 minus 119897

119903)2


1198872(1199062) =

1

21198882(1199062)2


(45)

1198873(119909) =

1

21198883(119909 minus 119897

119903)2

1198883isin (0infin) (46)


minus119905= minus

(119902119905)2

1198882

+ 1198883+ 1199021199052 (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+ 119902119905[minus119903119891minus119903119910

119905

119879

minus1

119905119910

119905+ (120590119890)2

+ (120590119894)2

] 1199021199051= 1198881

(47)

minus 119903

119905= minus

1198883(119909119903

119905minus 119897119903)

119902119905

minus 119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

minus [1199061

119905minus 119903119890

119905minus (120590119890)2

] minus (119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

)

minus (120590119894)2

119909119903(1199051) = 119897119903

(48)

minus 119897119905= minus119903119891119897119905+ 1198883(119909119903

119905minus 119897119903)2

minus 119902119905(120590119890)2

(119909119903

119905minus 1)2

minus 119902119905(120590119894)2

(119909119903

119905)2

119897 (1199051) = 0

(49)





1199062lowast

119905= minus

(119909 minus 119909119903

119905) 119902119905

1198882

1198922lowast

(119905 119909) = 1199062lowast

1198922lowast


lowast

119905= minus

(119909 minus 119909119903

119905)

119909minus1

119905119910

119905 1198923lowast

119879 timesX 997888rarr R119896

(50)

1198923119896lowast




(51)


V (119905 119909) = exp (minus119903119891(1199051minus 1199050)) [

1

2(119909 minus 119909

119903

119905)2

119902119905+

1

2119897119905] (52)

119869lowast= 119869 (119892

lowast) = E [V (119905

0 1199090)] (53)




minus119905= minus119887(119902

119905)2

+ 119888 + 2119886119905119902119905 1199021199051= 1198881

119887 =1

1198882 119888 = 119888

3isin (0infin) 119886 119879 997888rarr R

(54)


1199021

1199051= 119902 (119905

1minus 119905) 119886

1

119905= 119886 (119905

1minus 119905) 119902

1 [0 1199051minus 1199050] 997888rarr R

1

119905= minus (119905

1minus 119905) = minus119887(119902

1

119905)2

+ 119888 + 21198861

1199051199021

119905 1199021

1199050= 1198881

0 = 1

119905+ 119887(1199021

119905)2

minus 119888 minus 21198861

119905119902119905 1199021

0= 1198881

(55)


minus2

119905= minus119887(119902

2

119905)2

+ 119888 1199022

1199051= 1198881 119887 119888 isin (0infin) (56)


119905gt 0 for all 119905 isin 119879 = [0 119905

1minus 1199050]


1minus 1199050] 1199021119905

ge

1199022

119905gt 0 if


0 119887) ge (


119905

minus1198861

119905119887


(57)



(119886119905)2


1

1198882minus

1

1198882)

(58)


=


1minus 1199050]




1

1198882minus

1

1198882ge 0

(119886119905)2

le 1198884le (119888 minus 119888) (119887 minus 119887) = (119888

3minus 1198883) (

1

1198882minus

1

1198882)

(59)


1minus 1199050] 119902(1199051minus

119905) = 1199021

119905ge 1199022

119905gt 0




Note first that

V (119905 119909) = exp (minus119903119891(119905 minus 1199050)) [

1

2(119909 minus 119909

119903

119905)2

119902119905+

1

2119897119905]

119863119905V (119905 119909) = minus119903

119891V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))


119905) 119903

119905119902119905+

1

2(119909 minus 119909

119903

119905)2

119905+

1

2

119897119905]

119863119909V (119905 119909) = exp (minus119903

119891(119905 minus 1199050)) (119909 minus 119909

119903

119905) 119902119905

119863119909119909V (119905 119909) = exp (minus119903

119891(119905 minus 1199050)) 119902119905

(60)


So

0 = 119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))11986311990621198872(1199062)


119903

119905) 119902119905+ 11988821199062]

1199062lowast

= minus(119909 minus 119909

119903

119905) 119902119905

1198882

lowast= minus

(119909 minus 119909119903

119905) 119902119905

119909119902119905

minus1

119905119910

119905

(61)


1199021199051= 1198881 119909

119903(1199051) = 119897119903 119897 (119905

1) = 0

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050))

times [1

2(119909 minus 119909

119903(1199051))2

1199021199051+

1

2119897 (1199051)]

= exp (minus119903119891(1199051minus 1199050))

1

21198881(119909 minus 119897

119903)2

= exp (minus119903119891(1199051minus 1199050)) 1198871(119909)


exp (+119903119891(119905 minus 1199050))

times [119863119905V (119905 119909) +

1

2[(120590119890)2

(1 minus 119909)2

+(120590119894)2

(119909)2]119863119909119909V (119905 119909)

+ 119909 [119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]119863119909V (119905 119909)

+ [1199061

119905minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ 1199062lowast

119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050)) 1198872(1199062lowast

)

+ exp (minus119903119891(119905 minus 1199050)) 1198873(119909)

minus1

2

(119863119909V (119905 119909))

2

119863119909119909V (119905 119909)

119903119910

119905

119879

minus1

119905119910

119905]

= minus119903119891 1

2(119909 minus 119909

119903

119905)2

119902119905minus

1

2119903119891119897119905minus (119909 minus 119909

119903

119905) 119902119905119903

119905

+1

2(119909 minus 119909

119903

119905)2

119905+

1

2

119897119905

+1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2] 119902119905

+ (119909 minus 119909119903

119905) 119902119905[119909 (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+1199061

119905minus 119903119890

119905minus (120590119890)2

]

minus1

2

(119909 minus 119909119903

119905)2

(119902119905)2

1198882

+1

21198883(119909 minus 119897

119903)2

minus1

2(119909 minus 119909

119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

=1

2(119909)2[ minus 119903119891119902119905+ 119905+ (120590119890)2

119902119905+ (120590119894)2

119902119905

+ 2119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

minus(119902119905)2

1198882

+ 1198883minus 119902119905

119903119910

119905

119879

minus1

119905119910

119905]

+ 119909 [ + 119903119891119909119903

119905119902119905minus 119902119905119903

119905minus 119909119903

119905119905minus (120590119890)2

119902119905

minus 119909119903

119905119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ 119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] +119909119903

119905(119902119905)2

1198882

minus1198883119897119903+ 119909119903

119905119902119905

119903119910

119905

119879

minus1

119905119910

119905]

+1

2(119909)0[ minus 119903119891(119909119903

119905)2

119902119905+ 2119909119903

119905119902119905119903

119905+ (119909119903

119905)2

119905minus 119903119891119897119905

+ (120590119890)2

119902119905minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

]

minus(119909119903

119905)2

(119902119905)2

1198882

+ 1198883(119897119903)2

minus(119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905+ 119897119905]

(62)



minus 119902119905119903

119905minus 119909119903

119905119905+ 119903119891119909119903

119905119902119905minus 2(120590119890)2

119902119905

minus 119909119903

119905119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ 119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

]

+119909119903

119905(119902119905)2

1198882

minus 1198883119897119903+ 119909119903

119905119902119905

119903119910

119905

119879

minus1

119905119910

119905

= minus119902119905119903

119905

+ 119909119903

119905[minus

(119902119905)2

1198882

+ 1198883

+ 2119902119905(119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+119902119905((120590119890)2

+ (120590119894)2

minus 119903119891minus119903119910

119905

119879

minus1

119905119910

119905)]

minus 119909119903

119905119902119905[ (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

minus119903119891minus119903119910

119905

119879

minus1

119905119910

119905]

+ 119902 [minus(120590119890)2

+ (1199061

119905minus 119903119890

119905minus (120590119890)2

)] +119909119903

119905(119902119905)2

1198882

minus 1198883119897119903

= 119902119905[minus119903

119905+

1198883(119909119903

119905minus 119897119903)

119902119905

]

+ 119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ (119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

) + (120590119894)2

+ (1199061

119905minus 119903119890

119905minus (120590119890)2

)

= 0

(63)



119897119905minus 119903119891119897119905minus 119903119891(119909119903

119905)2

119902119905

+ 2119909119903

119905[119902119905119903

119905+ 119909119903

119905119905] minus (119909

119903

119905)2

119905+ (120590119890)2

119902119905

minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] + 1198883(119897119903)2

minus(119909119903

119905)2

(119902119905)2

1198882

minus (119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905


119897119905minus 119903119891119897119905minus 119903119891(119909119903

119905)2

119902119905+ (120590119890)2

119902119905

minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] + 1198883(119897119903)2

minus(119909119903

119905)2

(119902119905)2

1198882

minus (119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

+ 2119903119891(119909119903

119905)2

119902119905minus 2119909119903

119905119902119905(120590119890)2

minus 2(119909119903

119905)2

119902119905[119903

T+ 119903119890minus 119903119894+ (120590119890)2

+ (120590119894)2

]

+ 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] minus 11988831198971199032119909119903

119905

+2(119909119903

119905)2

(119902119905)2

1198882

+ 2(119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

minus(119909119903

119905)2

(119902119905)2

1198882

+ 1198883(119909119903

119905)2

+ (119909119903

119905)2

1199021199052 (119903

T+ 119903119890minus 119903119894+ (120590119890)2

+ (120590119894)2

)

+ (119909119903

119905)2

119902119905(minus119903119891+ (120590119890)2

+ (120590119894)2

minus119903119910

119905

119879

minus1

119905119910

119905)

= 119897119905minus 119903119891119897119905+ 1198883(119909119903

119905minus 119897119903)2

minus (120590119890)2

119902119905(119909119903

119905minus 1)2

minus (120590119894)2

119902119905(119909119903

119905)2

= 0

(64)







2gt 0 should


2(1199062) = (12)119888

2(1199062)2 given by (45) This







3(119909) = (12)119888



1198873(119909) = 119888

3[exp (119909 minus 119897

119903) + (119897119903minus 119909) minus 1] (65)







1199051198882 which





119903(119909) = (12)119888

3(119909 minus 119909




119903



119905119910



minus1

119905119910


deviation (119909 minus 119909119903

119905)




1199061

119905minus 119903119890

119905minus (120590119890)2

(66)



119890

119905is the


119903119890



1199051 119909119905lt 119897119903The term 119888

3(119909119903





119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

] (67)




T(119905) + (120590

119890)2+ (120590119894)2 is the effect of


(119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

) minus (120590119894)2

(68)




minus119903

119905= minus119896 (119909

119903

119905minus 119898) 119909

119903(1199051) = 119897119903

119896 = (119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2

+ (120590119894)2

)) +1198883

1199020

119898 =


) + (120590119890)2

(119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2+ (120590119894)2

)) + 11988831199020

(69)


119902119905

= 1199020 and lim

119905darr0119909119903




1199062lowast

119905= minus

(119909119905minus 119898) 119902

0

1198882

= gt 0 if 119909

119905lt 119898

lt 0 if 119909119905gt 119898

120587lowast

119905= minus

(119909119905minus 119898)

119909119905

119895

= gt 0 if 119909

119905lt 119898


(70)


Case 1 (119909119905











119898 =


) + (120590119890)2

(119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2+ (120590119894)2

)) + 11988831199020

(71)





Appendices













References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014








5



50











119889119909119905= 119860119905119909119905119889119905 +

119899

sum

119895=0

119861119895119909119905119892119895(119905 119909119905) 119889119905 + 119886

119905119889119905

+

3

sum

119895=1

119872119895119895(119892 (119905 119909

119905)) 119909119905119889119882119895119895

119905 119909 (119905

0) = 1199090

(23)



119869 (119892) = E [int

1199051

1199050

exp (minus119903119891(119897 minus 1199050)) 119887 (119897 119909

119897 119892 (119897 119909

119897)) 119889119897

+ exp (minus119903119891(1199051minus 1199050)) 1198871(119909 (1199051)) ]

(24)


where 119892 isin G119860 119879 = [119905

0 1199051] and 119887



119887 (119905 119909 119906) = 1198872(1199062) + 1198873(1199091

1199092) (25)

for 1198872 U2rarr R+ and 119887





= 0







119869 (119892) (26)



119869 (119892) (27)



119892isinG119860

119869 (119892) isin G119860 (28)





119869 (119892)


119869 (119892)

(29)

119869 (119892) = E [int

1199051

1199050

exp (minus119903119891(119897 minus 1199050)) [1198872(1199062

119905) + 1198873(119909119905)] d119905

+ exp (minus119903119891(1199051minus 1199050)) 1198871(119909 (1199051)) ]

(30)


3 R+ rarr R+





1198872(1199062) isin 1198622(R)


11986311990621198872(1199062) = minusinfin


11986311990621198872(1199062) = +infin

119863119906211990621198872(1199062) gt 0 forall119906

2isin R

(31)


2



0 = 119863119905V (119905 119909) +

1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909119905)2

]119863119909119909V (119905 119909)

+ 119909 (119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)119863119909V (119905 119909)

+ [1199061

119905minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ 1199062

119905

lowast

119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050)) 1198872(1199062

119905

lowast

)


[119863119909V (119905 119909)]

2

2119863119909119909V (119905 119909)

119903119910

119905

119879

minus1

119905119910

119905

(32)

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050)) 1198871(119909) (33)


0 = 119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))11986311990621198872(1199062

119905) (34)


1198922lowast

(119905 119909) = 1199062lowast

1198922lowast


with 1199062lowast


lowast= minus

119863119909V (119905 119909)

119909119863119909119909V (119905 119909)

minus1

119905119910

119905

1198923119896lowast



(36)



119869lowast= 119869 (119892

lowast) = E [V (119905 119909

0)] (37)



12(119879timesX)

is given by

0 = 119863119905119908 (119905 119909)


[1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2

+(119909)2119879119905]119863119909119909119908 (119905 119909)

+ [ (119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

+119903119910

119905

119879

) 119909]119863119909119908 (119905 119909)

+ [1199061

119905+ 1199062minus 119903119890

119905minus (120590119890)2

]119863119909119908 (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) [1198872(1199062) + 1198873(119909)] ]

119908 (1199051 119909) = exp (minus119903

119891(1199051minus 1199050)) 1198871(119909)

(38)



(b) depending on




inf1199062isinR

1198672(119905 119909 119906

2)

1198672(119905 119909 119906) = 119906

2119863119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) 1198872(1199062)

11986311990621198672(119905 119909 119906) = 119863

119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050))11986311990621198872(1199062) = 0

(39)


2lowast

isin U = R

and

inf1199062isinR

1198672(119905 119909 119906

2) = 119867

2(119905 119909 119906

2lowast

)

= 1199062lowast

119863119909V (119905 119909)


)

(40)




infisinR119896

1198673(119905 119909 )

1198673(119905 119909 ) =

1

2[119879

119905119905119905] (119909)2119863119909119909V (119905 119909)

+119903119910

119905

119879

119909119863119909V (119905 119909)

=1

2(119905+ (

119909 [119863119909V (119905 119909)]

(119909)2119863119909119909V (119905 119909)

) minus1

119905119910

119905)

119879

times 119905(119909)2119863119909119909V (119905 119909)

times (119905+ (

119909 [119863119909V (119905 119909)]

(119909)2119863119909119909V (119905 119909)

) minus1

119905119910

119905)

minus1

2(

[119909119863119909V (119905 119909)]

2

(119909)2119863119909119909V (119905 119909)

)119903119910

119905

119879

minus1

119905119910

119905

119905(119909)2119863119909119909V (119905 119909) gt 0 forall119909 isin R 0

(41)

Hence we have that

lowast= minus

119863119909V (119905 119909)

119909119863119909119909V (119905 119909)

minus1

119905119910

119905

1198923119896lowast

(119905 119909) =

119896lowast if

119896

sum

119894=1

119894lowast

isin [0 1]

119896lowast

sum119896

119895=1119895lowast

if119896

sum

119894=1

119894lowast

gt 1


1198673(119905 119909

lowast) = minus

[119863119909V (119905 119909)]

2

2119863119909119909V (119905 119909)

(119910

119905)119879

minus1

119905119910

119905

(42)





0 = 119863119905V (119905 119909)


1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2

+(119909)2(119879119905) ]119863

119909119909V (119905 119909)

+ [119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+(120590119894)2

+119903119910

119905

119879

] 119909

+ [1199061

119905+ 1199062minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) [1198872(1199062) + 1198873(119909)]

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050)) 1198871(119909)

(43)



lowast=

119869(119892lowast) = E[V(119905

0 1199090)]



119869 (119892) = E [int

1199051

1199050

exp (minus119903119891(119897 minus 1199050))

times [1

21198882((1199062)2

(119897)) +1

21198883(119909119905minus 119897119903)2

] 119889119897

+1

21198881(119909 (1199051) minus 119897119903)2 exp (minus119903

119891(1199051minus 1199050)) ]

(44)


1198871(119909) =

1

21198881(119909 minus 119897

119903)2


1198872(1199062) =

1

21198882(1199062)2


(45)

1198873(119909) =

1

21198883(119909 minus 119897

119903)2

1198883isin (0infin) (46)


minus119905= minus

(119902119905)2

1198882

+ 1198883+ 1199021199052 (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+ 119902119905[minus119903119891minus119903119910

119905

119879

minus1

119905119910

119905+ (120590119890)2

+ (120590119894)2

] 1199021199051= 1198881

(47)

minus 119903

119905= minus

1198883(119909119903

119905minus 119897119903)

119902119905

minus 119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

minus [1199061

119905minus 119903119890

119905minus (120590119890)2

] minus (119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

)

minus (120590119894)2

119909119903(1199051) = 119897119903

(48)

minus 119897119905= minus119903119891119897119905+ 1198883(119909119903

119905minus 119897119903)2

minus 119902119905(120590119890)2

(119909119903

119905minus 1)2

minus 119902119905(120590119894)2

(119909119903

119905)2

119897 (1199051) = 0

(49)





1199062lowast

119905= minus

(119909 minus 119909119903

119905) 119902119905

1198882

1198922lowast

(119905 119909) = 1199062lowast

1198922lowast


lowast

119905= minus

(119909 minus 119909119903

119905)

119909minus1

119905119910

119905 1198923lowast

119879 timesX 997888rarr R119896

(50)

1198923119896lowast




(51)


V (119905 119909) = exp (minus119903119891(1199051minus 1199050)) [

1

2(119909 minus 119909

119903

119905)2

119902119905+

1

2119897119905] (52)

119869lowast= 119869 (119892

lowast) = E [V (119905

0 1199090)] (53)




minus119905= minus119887(119902

119905)2

+ 119888 + 2119886119905119902119905 1199021199051= 1198881

119887 =1

1198882 119888 = 119888

3isin (0infin) 119886 119879 997888rarr R

(54)


1199021

1199051= 119902 (119905

1minus 119905) 119886

1

119905= 119886 (119905

1minus 119905) 119902

1 [0 1199051minus 1199050] 997888rarr R

1

119905= minus (119905

1minus 119905) = minus119887(119902

1

119905)2

+ 119888 + 21198861

1199051199021

119905 1199021

1199050= 1198881

0 = 1

119905+ 119887(1199021

119905)2

minus 119888 minus 21198861

119905119902119905 1199021

0= 1198881

(55)


minus2

119905= minus119887(119902

2

119905)2

+ 119888 1199022

1199051= 1198881 119887 119888 isin (0infin) (56)


119905gt 0 for all 119905 isin 119879 = [0 119905

1minus 1199050]


1minus 1199050] 1199021119905

ge

1199022

119905gt 0 if


0 119887) ge (


119905

minus1198861

119905119887


(57)



(119886119905)2


1

1198882minus

1

1198882)

(58)


=


1minus 1199050]




1

1198882minus

1

1198882ge 0

(119886119905)2

le 1198884le (119888 minus 119888) (119887 minus 119887) = (119888

3minus 1198883) (

1

1198882minus

1

1198882)

(59)


1minus 1199050] 119902(1199051minus

119905) = 1199021

119905ge 1199022

119905gt 0




Note first that

V (119905 119909) = exp (minus119903119891(119905 minus 1199050)) [

1

2(119909 minus 119909

119903

119905)2

119902119905+

1

2119897119905]

119863119905V (119905 119909) = minus119903

119891V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))


119905) 119903

119905119902119905+

1

2(119909 minus 119909

119903

119905)2

119905+

1

2

119897119905]

119863119909V (119905 119909) = exp (minus119903

119891(119905 minus 1199050)) (119909 minus 119909

119903

119905) 119902119905

119863119909119909V (119905 119909) = exp (minus119903

119891(119905 minus 1199050)) 119902119905

(60)


So

0 = 119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))11986311990621198872(1199062)


119903

119905) 119902119905+ 11988821199062]

1199062lowast

= minus(119909 minus 119909

119903

119905) 119902119905

1198882

lowast= minus

(119909 minus 119909119903

119905) 119902119905

119909119902119905

minus1

119905119910

119905

(61)


1199021199051= 1198881 119909

119903(1199051) = 119897119903 119897 (119905

1) = 0

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050))

times [1

2(119909 minus 119909

119903(1199051))2

1199021199051+

1

2119897 (1199051)]

= exp (minus119903119891(1199051minus 1199050))

1

21198881(119909 minus 119897

119903)2

= exp (minus119903119891(1199051minus 1199050)) 1198871(119909)


exp (+119903119891(119905 minus 1199050))

times [119863119905V (119905 119909) +

1

2[(120590119890)2

(1 minus 119909)2

+(120590119894)2

(119909)2]119863119909119909V (119905 119909)

+ 119909 [119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]119863119909V (119905 119909)

+ [1199061

119905minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ 1199062lowast

119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050)) 1198872(1199062lowast

)

+ exp (minus119903119891(119905 minus 1199050)) 1198873(119909)

minus1

2

(119863119909V (119905 119909))

2

119863119909119909V (119905 119909)

119903119910

119905

119879

minus1

119905119910

119905]

= minus119903119891 1

2(119909 minus 119909

119903

119905)2

119902119905minus

1

2119903119891119897119905minus (119909 minus 119909

119903

119905) 119902119905119903

119905

+1

2(119909 minus 119909

119903

119905)2

119905+

1

2

119897119905

+1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2] 119902119905

+ (119909 minus 119909119903

119905) 119902119905[119909 (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+1199061

119905minus 119903119890

119905minus (120590119890)2

]

minus1

2

(119909 minus 119909119903

119905)2

(119902119905)2

1198882

+1

21198883(119909 minus 119897

119903)2

minus1

2(119909 minus 119909

119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

=1

2(119909)2[ minus 119903119891119902119905+ 119905+ (120590119890)2

119902119905+ (120590119894)2

119902119905

+ 2119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

minus(119902119905)2

1198882

+ 1198883minus 119902119905

119903119910

119905

119879

minus1

119905119910

119905]

+ 119909 [ + 119903119891119909119903

119905119902119905minus 119902119905119903

119905minus 119909119903

119905119905minus (120590119890)2

119902119905

minus 119909119903

119905119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ 119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] +119909119903

119905(119902119905)2

1198882

minus1198883119897119903+ 119909119903

119905119902119905

119903119910

119905

119879

minus1

119905119910

119905]

+1

2(119909)0[ minus 119903119891(119909119903

119905)2

119902119905+ 2119909119903

119905119902119905119903

119905+ (119909119903

119905)2

119905minus 119903119891119897119905

+ (120590119890)2

119902119905minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

]

minus(119909119903

119905)2

(119902119905)2

1198882

+ 1198883(119897119903)2

minus(119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905+ 119897119905]

(62)



minus 119902119905119903

119905minus 119909119903

119905119905+ 119903119891119909119903

119905119902119905minus 2(120590119890)2

119902119905

minus 119909119903

119905119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ 119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

]

+119909119903

119905(119902119905)2

1198882

minus 1198883119897119903+ 119909119903

119905119902119905

119903119910

119905

119879

minus1

119905119910

119905

= minus119902119905119903

119905

+ 119909119903

119905[minus

(119902119905)2

1198882

+ 1198883

+ 2119902119905(119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+119902119905((120590119890)2

+ (120590119894)2

minus 119903119891minus119903119910

119905

119879

minus1

119905119910

119905)]

minus 119909119903

119905119902119905[ (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

minus119903119891minus119903119910

119905

119879

minus1

119905119910

119905]

+ 119902 [minus(120590119890)2

+ (1199061

119905minus 119903119890

119905minus (120590119890)2

)] +119909119903

119905(119902119905)2

1198882

minus 1198883119897119903

= 119902119905[minus119903

119905+

1198883(119909119903

119905minus 119897119903)

119902119905

]

+ 119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ (119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

) + (120590119894)2

+ (1199061

119905minus 119903119890

119905minus (120590119890)2

)

= 0

(63)



119897119905minus 119903119891119897119905minus 119903119891(119909119903

119905)2

119902119905

+ 2119909119903

119905[119902119905119903

119905+ 119909119903

119905119905] minus (119909

119903

119905)2

119905+ (120590119890)2

119902119905

minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] + 1198883(119897119903)2

minus(119909119903

119905)2

(119902119905)2

1198882

minus (119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905


119897119905minus 119903119891119897119905minus 119903119891(119909119903

119905)2

119902119905+ (120590119890)2

119902119905

minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] + 1198883(119897119903)2

minus(119909119903

119905)2

(119902119905)2

1198882

minus (119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

+ 2119903119891(119909119903

119905)2

119902119905minus 2119909119903

119905119902119905(120590119890)2

minus 2(119909119903

119905)2

119902119905[119903

T+ 119903119890minus 119903119894+ (120590119890)2

+ (120590119894)2

]

+ 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] minus 11988831198971199032119909119903

119905

+2(119909119903

119905)2

(119902119905)2

1198882

+ 2(119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

minus(119909119903

119905)2

(119902119905)2

1198882

+ 1198883(119909119903

119905)2

+ (119909119903

119905)2

1199021199052 (119903

T+ 119903119890minus 119903119894+ (120590119890)2

+ (120590119894)2

)

+ (119909119903

119905)2

119902119905(minus119903119891+ (120590119890)2

+ (120590119894)2

minus119903119910

119905

119879

minus1

119905119910

119905)

= 119897119905minus 119903119891119897119905+ 1198883(119909119903

119905minus 119897119903)2

minus (120590119890)2

119902119905(119909119903

119905minus 1)2

minus (120590119894)2

119902119905(119909119903

119905)2

= 0

(64)







2gt 0 should


2(1199062) = (12)119888

2(1199062)2 given by (45) This







3(119909) = (12)119888



1198873(119909) = 119888

3[exp (119909 minus 119897

119903) + (119897119903minus 119909) minus 1] (65)







1199051198882 which





119903(119909) = (12)119888

3(119909 minus 119909




119903



119905119910



minus1

119905119910


deviation (119909 minus 119909119903

119905)




1199061

119905minus 119903119890

119905minus (120590119890)2

(66)



119890

119905is the


119903119890



1199051 119909119905lt 119897119903The term 119888

3(119909119903





119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

] (67)




T(119905) + (120590

119890)2+ (120590119894)2 is the effect of


(119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

) minus (120590119894)2

(68)




minus119903

119905= minus119896 (119909

119903

119905minus 119898) 119909

119903(1199051) = 119897119903

119896 = (119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2

+ (120590119894)2

)) +1198883

1199020

119898 =


) + (120590119890)2

(119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2+ (120590119894)2

)) + 11988831199020

(69)


119902119905

= 1199020 and lim

119905darr0119909119903




1199062lowast

119905= minus

(119909119905minus 119898) 119902

0

1198882

= gt 0 if 119909

119905lt 119898

lt 0 if 119909119905gt 119898

120587lowast

119905= minus

(119909119905minus 119898)

119909119905

119895

= gt 0 if 119909

119905lt 119898


(70)


Case 1 (119909119905











119898 =


) + (120590119890)2

(119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2+ (120590119894)2

)) + 11988831199020

(71)





Appendices













References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




where 119892 isin G119860 119879 = [119905

0 1199051] and 119887



119887 (119905 119909 119906) = 1198872(1199062) + 1198873(1199091

1199092) (25)

for 1198872 U2rarr R+ and 119887





= 0







119869 (119892) (26)



119869 (119892) (27)



119892isinG119860

119869 (119892) isin G119860 (28)





119869 (119892)


119869 (119892)

(29)

119869 (119892) = E [int

1199051

1199050

exp (minus119903119891(119897 minus 1199050)) [1198872(1199062

119905) + 1198873(119909119905)] d119905

+ exp (minus119903119891(1199051minus 1199050)) 1198871(119909 (1199051)) ]

(30)


3 R+ rarr R+





1198872(1199062) isin 1198622(R)


11986311990621198872(1199062) = minusinfin


11986311990621198872(1199062) = +infin

119863119906211990621198872(1199062) gt 0 forall119906

2isin R

(31)


2



0 = 119863119905V (119905 119909) +

1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909119905)2

]119863119909119909V (119905 119909)

+ 119909 (119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)119863119909V (119905 119909)

+ [1199061

119905minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ 1199062

119905

lowast

119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050)) 1198872(1199062

119905

lowast

)


[119863119909V (119905 119909)]

2

2119863119909119909V (119905 119909)

119903119910

119905

119879

minus1

119905119910

119905

(32)

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050)) 1198871(119909) (33)


0 = 119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))11986311990621198872(1199062

119905) (34)


1198922lowast

(119905 119909) = 1199062lowast

1198922lowast


with 1199062lowast


lowast= minus

119863119909V (119905 119909)

119909119863119909119909V (119905 119909)

minus1

119905119910

119905

1198923119896lowast



(36)



119869lowast= 119869 (119892

lowast) = E [V (119905 119909

0)] (37)



12(119879timesX)

is given by

0 = 119863119905119908 (119905 119909)


[1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2

+(119909)2119879119905]119863119909119909119908 (119905 119909)

+ [ (119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

+119903119910

119905

119879

) 119909]119863119909119908 (119905 119909)

+ [1199061

119905+ 1199062minus 119903119890

119905minus (120590119890)2

]119863119909119908 (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) [1198872(1199062) + 1198873(119909)] ]

119908 (1199051 119909) = exp (minus119903

119891(1199051minus 1199050)) 1198871(119909)

(38)



(b) depending on




inf1199062isinR

1198672(119905 119909 119906

2)

1198672(119905 119909 119906) = 119906

2119863119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) 1198872(1199062)

11986311990621198672(119905 119909 119906) = 119863

119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050))11986311990621198872(1199062) = 0

(39)


2lowast

isin U = R

and

inf1199062isinR

1198672(119905 119909 119906

2) = 119867

2(119905 119909 119906

2lowast

)

= 1199062lowast

119863119909V (119905 119909)


)

(40)




infisinR119896

1198673(119905 119909 )

1198673(119905 119909 ) =

1

2[119879

119905119905119905] (119909)2119863119909119909V (119905 119909)

+119903119910

119905

119879

119909119863119909V (119905 119909)

=1

2(119905+ (

119909 [119863119909V (119905 119909)]

(119909)2119863119909119909V (119905 119909)

) minus1

119905119910

119905)

119879

times 119905(119909)2119863119909119909V (119905 119909)

times (119905+ (

119909 [119863119909V (119905 119909)]

(119909)2119863119909119909V (119905 119909)

) minus1

119905119910

119905)

minus1

2(

[119909119863119909V (119905 119909)]

2

(119909)2119863119909119909V (119905 119909)

)119903119910

119905

119879

minus1

119905119910

119905

119905(119909)2119863119909119909V (119905 119909) gt 0 forall119909 isin R 0

(41)

Hence we have that

lowast= minus

119863119909V (119905 119909)

119909119863119909119909V (119905 119909)

minus1

119905119910

119905

1198923119896lowast

(119905 119909) =

119896lowast if

119896

sum

119894=1

119894lowast

isin [0 1]

119896lowast

sum119896

119895=1119895lowast

if119896

sum

119894=1

119894lowast

gt 1


1198673(119905 119909

lowast) = minus

[119863119909V (119905 119909)]

2

2119863119909119909V (119905 119909)

(119910

119905)119879

minus1

119905119910

119905

(42)





0 = 119863119905V (119905 119909)


1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2

+(119909)2(119879119905) ]119863

119909119909V (119905 119909)

+ [119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+(120590119894)2

+119903119910

119905

119879

] 119909

+ [1199061

119905+ 1199062minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) [1198872(1199062) + 1198873(119909)]

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050)) 1198871(119909)

(43)



lowast=

119869(119892lowast) = E[V(119905

0 1199090)]



119869 (119892) = E [int

1199051

1199050

exp (minus119903119891(119897 minus 1199050))

times [1

21198882((1199062)2

(119897)) +1

21198883(119909119905minus 119897119903)2

] 119889119897

+1

21198881(119909 (1199051) minus 119897119903)2 exp (minus119903

119891(1199051minus 1199050)) ]

(44)


1198871(119909) =

1

21198881(119909 minus 119897

119903)2


1198872(1199062) =

1

21198882(1199062)2


(45)

1198873(119909) =

1

21198883(119909 minus 119897

119903)2

1198883isin (0infin) (46)


minus119905= minus

(119902119905)2

1198882

+ 1198883+ 1199021199052 (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+ 119902119905[minus119903119891minus119903119910

119905

119879

minus1

119905119910

119905+ (120590119890)2

+ (120590119894)2

] 1199021199051= 1198881

(47)

minus 119903

119905= minus

1198883(119909119903

119905minus 119897119903)

119902119905

minus 119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

minus [1199061

119905minus 119903119890

119905minus (120590119890)2

] minus (119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

)

minus (120590119894)2

119909119903(1199051) = 119897119903

(48)

minus 119897119905= minus119903119891119897119905+ 1198883(119909119903

119905minus 119897119903)2

minus 119902119905(120590119890)2

(119909119903

119905minus 1)2

minus 119902119905(120590119894)2

(119909119903

119905)2

119897 (1199051) = 0

(49)





1199062lowast

119905= minus

(119909 minus 119909119903

119905) 119902119905

1198882

1198922lowast

(119905 119909) = 1199062lowast

1198922lowast


lowast

119905= minus

(119909 minus 119909119903

119905)

119909minus1

119905119910

119905 1198923lowast

119879 timesX 997888rarr R119896

(50)

1198923119896lowast




(51)


V (119905 119909) = exp (minus119903119891(1199051minus 1199050)) [

1

2(119909 minus 119909

119903

119905)2

119902119905+

1

2119897119905] (52)

119869lowast= 119869 (119892

lowast) = E [V (119905

0 1199090)] (53)




minus119905= minus119887(119902

119905)2

+ 119888 + 2119886119905119902119905 1199021199051= 1198881

119887 =1

1198882 119888 = 119888

3isin (0infin) 119886 119879 997888rarr R

(54)


1199021

1199051= 119902 (119905

1minus 119905) 119886

1

119905= 119886 (119905

1minus 119905) 119902

1 [0 1199051minus 1199050] 997888rarr R

1

119905= minus (119905

1minus 119905) = minus119887(119902

1

119905)2

+ 119888 + 21198861

1199051199021

119905 1199021

1199050= 1198881

0 = 1

119905+ 119887(1199021

119905)2

minus 119888 minus 21198861

119905119902119905 1199021

0= 1198881

(55)


minus2

119905= minus119887(119902

2

119905)2

+ 119888 1199022

1199051= 1198881 119887 119888 isin (0infin) (56)


119905gt 0 for all 119905 isin 119879 = [0 119905

1minus 1199050]


1minus 1199050] 1199021119905

ge

1199022

119905gt 0 if


0 119887) ge (


119905

minus1198861

119905119887


(57)



(119886119905)2


1

1198882minus

1

1198882)

(58)


=


1minus 1199050]




1

1198882minus

1

1198882ge 0

(119886119905)2

le 1198884le (119888 minus 119888) (119887 minus 119887) = (119888

3minus 1198883) (

1

1198882minus

1

1198882)

(59)


1minus 1199050] 119902(1199051minus

119905) = 1199021

119905ge 1199022

119905gt 0




Note first that

V (119905 119909) = exp (minus119903119891(119905 minus 1199050)) [

1

2(119909 minus 119909

119903

119905)2

119902119905+

1

2119897119905]

119863119905V (119905 119909) = minus119903

119891V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))


119905) 119903

119905119902119905+

1

2(119909 minus 119909

119903

119905)2

119905+

1

2

119897119905]

119863119909V (119905 119909) = exp (minus119903

119891(119905 minus 1199050)) (119909 minus 119909

119903

119905) 119902119905

119863119909119909V (119905 119909) = exp (minus119903

119891(119905 minus 1199050)) 119902119905

(60)


So

0 = 119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))11986311990621198872(1199062)


119903

119905) 119902119905+ 11988821199062]

1199062lowast

= minus(119909 minus 119909

119903

119905) 119902119905

1198882

lowast= minus

(119909 minus 119909119903

119905) 119902119905

119909119902119905

minus1

119905119910

119905

(61)


1199021199051= 1198881 119909

119903(1199051) = 119897119903 119897 (119905

1) = 0

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050))

times [1

2(119909 minus 119909

119903(1199051))2

1199021199051+

1

2119897 (1199051)]

= exp (minus119903119891(1199051minus 1199050))

1

21198881(119909 minus 119897

119903)2

= exp (minus119903119891(1199051minus 1199050)) 1198871(119909)


exp (+119903119891(119905 minus 1199050))

times [119863119905V (119905 119909) +

1

2[(120590119890)2

(1 minus 119909)2

+(120590119894)2

(119909)2]119863119909119909V (119905 119909)

+ 119909 [119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]119863119909V (119905 119909)

+ [1199061

119905minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ 1199062lowast

119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050)) 1198872(1199062lowast

)

+ exp (minus119903119891(119905 minus 1199050)) 1198873(119909)

minus1

2

(119863119909V (119905 119909))

2

119863119909119909V (119905 119909)

119903119910

119905

119879

minus1

119905119910

119905]

= minus119903119891 1

2(119909 minus 119909

119903

119905)2

119902119905minus

1

2119903119891119897119905minus (119909 minus 119909

119903

119905) 119902119905119903

119905

+1

2(119909 minus 119909

119903

119905)2

119905+

1

2

119897119905

+1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2] 119902119905

+ (119909 minus 119909119903

119905) 119902119905[119909 (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+1199061

119905minus 119903119890

119905minus (120590119890)2

]

minus1

2

(119909 minus 119909119903

119905)2

(119902119905)2

1198882

+1

21198883(119909 minus 119897

119903)2

minus1

2(119909 minus 119909

119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

=1

2(119909)2[ minus 119903119891119902119905+ 119905+ (120590119890)2

119902119905+ (120590119894)2

119902119905

+ 2119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

minus(119902119905)2

1198882

+ 1198883minus 119902119905

119903119910

119905

119879

minus1

119905119910

119905]

+ 119909 [ + 119903119891119909119903

119905119902119905minus 119902119905119903

119905minus 119909119903

119905119905minus (120590119890)2

119902119905

minus 119909119903

119905119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ 119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] +119909119903

119905(119902119905)2

1198882

minus1198883119897119903+ 119909119903

119905119902119905

119903119910

119905

119879

minus1

119905119910

119905]

+1

2(119909)0[ minus 119903119891(119909119903

119905)2

119902119905+ 2119909119903

119905119902119905119903

119905+ (119909119903

119905)2

119905minus 119903119891119897119905

+ (120590119890)2

119902119905minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

]

minus(119909119903

119905)2

(119902119905)2

1198882

+ 1198883(119897119903)2

minus(119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905+ 119897119905]

(62)



minus 119902119905119903

119905minus 119909119903

119905119905+ 119903119891119909119903

119905119902119905minus 2(120590119890)2

119902119905

minus 119909119903

119905119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ 119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

]

+119909119903

119905(119902119905)2

1198882

minus 1198883119897119903+ 119909119903

119905119902119905

119903119910

119905

119879

minus1

119905119910

119905

= minus119902119905119903

119905

+ 119909119903

119905[minus

(119902119905)2

1198882

+ 1198883

+ 2119902119905(119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+119902119905((120590119890)2

+ (120590119894)2

minus 119903119891minus119903119910

119905

119879

minus1

119905119910

119905)]

minus 119909119903

119905119902119905[ (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

minus119903119891minus119903119910

119905

119879

minus1

119905119910

119905]

+ 119902 [minus(120590119890)2

+ (1199061

119905minus 119903119890

119905minus (120590119890)2

)] +119909119903

119905(119902119905)2

1198882

minus 1198883119897119903

= 119902119905[minus119903

119905+

1198883(119909119903

119905minus 119897119903)

119902119905

]

+ 119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ (119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

) + (120590119894)2

+ (1199061

119905minus 119903119890

119905minus (120590119890)2

)

= 0

(63)



119897119905minus 119903119891119897119905minus 119903119891(119909119903

119905)2

119902119905

+ 2119909119903

119905[119902119905119903

119905+ 119909119903

119905119905] minus (119909

119903

119905)2

119905+ (120590119890)2

119902119905

minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] + 1198883(119897119903)2

minus(119909119903

119905)2

(119902119905)2

1198882

minus (119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905


119897119905minus 119903119891119897119905minus 119903119891(119909119903

119905)2

119902119905+ (120590119890)2

119902119905

minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] + 1198883(119897119903)2

minus(119909119903

119905)2

(119902119905)2

1198882

minus (119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

+ 2119903119891(119909119903

119905)2

119902119905minus 2119909119903

119905119902119905(120590119890)2

minus 2(119909119903

119905)2

119902119905[119903

T+ 119903119890minus 119903119894+ (120590119890)2

+ (120590119894)2

]

+ 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] minus 11988831198971199032119909119903

119905

+2(119909119903

119905)2

(119902119905)2

1198882

+ 2(119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

minus(119909119903

119905)2

(119902119905)2

1198882

+ 1198883(119909119903

119905)2

+ (119909119903

119905)2

1199021199052 (119903

T+ 119903119890minus 119903119894+ (120590119890)2

+ (120590119894)2

)

+ (119909119903

119905)2

119902119905(minus119903119891+ (120590119890)2

+ (120590119894)2

minus119903119910

119905

119879

minus1

119905119910

119905)

= 119897119905minus 119903119891119897119905+ 1198883(119909119903

119905minus 119897119903)2

minus (120590119890)2

119902119905(119909119903

119905minus 1)2

minus (120590119894)2

119902119905(119909119903

119905)2

= 0

(64)







2gt 0 should


2(1199062) = (12)119888

2(1199062)2 given by (45) This







3(119909) = (12)119888



1198873(119909) = 119888

3[exp (119909 minus 119897

119903) + (119897119903minus 119909) minus 1] (65)







1199051198882 which





119903(119909) = (12)119888

3(119909 minus 119909




119903



119905119910



minus1

119905119910


deviation (119909 minus 119909119903

119905)




1199061

119905minus 119903119890

119905minus (120590119890)2

(66)



119890

119905is the


119903119890



1199051 119909119905lt 119897119903The term 119888

3(119909119903





119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

] (67)




T(119905) + (120590

119890)2+ (120590119894)2 is the effect of


(119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

) minus (120590119894)2

(68)




minus119903

119905= minus119896 (119909

119903

119905minus 119898) 119909

119903(1199051) = 119897119903

119896 = (119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2

+ (120590119894)2

)) +1198883

1199020

119898 =


) + (120590119890)2

(119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2+ (120590119894)2

)) + 11988831199020

(69)


119902119905

= 1199020 and lim

119905darr0119909119903




1199062lowast

119905= minus

(119909119905minus 119898) 119902

0

1198882

= gt 0 if 119909

119905lt 119898

lt 0 if 119909119905gt 119898

120587lowast

119905= minus

(119909119905minus 119898)

119909119905

119895

= gt 0 if 119909

119905lt 119898


(70)


Case 1 (119909119905











119898 =


) + (120590119890)2

(119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2+ (120590119894)2

)) + 11988831199020

(71)





Appendices













References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





119869lowast= 119869 (119892

lowast) = E [V (119905 119909

0)] (37)



12(119879timesX)

is given by

0 = 119863119905119908 (119905 119909)


[1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2

+(119909)2119879119905]119863119909119909119908 (119905 119909)

+ [ (119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

+119903119910

119905

119879

) 119909]119863119909119908 (119905 119909)

+ [1199061

119905+ 1199062minus 119903119890

119905minus (120590119890)2

]119863119909119908 (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) [1198872(1199062) + 1198873(119909)] ]

119908 (1199051 119909) = exp (minus119903

119891(1199051minus 1199050)) 1198871(119909)

(38)



(b) depending on




inf1199062isinR

1198672(119905 119909 119906

2)

1198672(119905 119909 119906) = 119906

2119863119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) 1198872(1199062)

11986311990621198672(119905 119909 119906) = 119863

119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050))11986311990621198872(1199062) = 0

(39)


2lowast

isin U = R

and

inf1199062isinR

1198672(119905 119909 119906

2) = 119867

2(119905 119909 119906

2lowast

)

= 1199062lowast

119863119909V (119905 119909)


)

(40)




infisinR119896

1198673(119905 119909 )

1198673(119905 119909 ) =

1

2[119879

119905119905119905] (119909)2119863119909119909V (119905 119909)

+119903119910

119905

119879

119909119863119909V (119905 119909)

=1

2(119905+ (

119909 [119863119909V (119905 119909)]

(119909)2119863119909119909V (119905 119909)

) minus1

119905119910

119905)

119879

times 119905(119909)2119863119909119909V (119905 119909)

times (119905+ (

119909 [119863119909V (119905 119909)]

(119909)2119863119909119909V (119905 119909)

) minus1

119905119910

119905)

minus1

2(

[119909119863119909V (119905 119909)]

2

(119909)2119863119909119909V (119905 119909)

)119903119910

119905

119879

minus1

119905119910

119905

119905(119909)2119863119909119909V (119905 119909) gt 0 forall119909 isin R 0

(41)

Hence we have that

lowast= minus

119863119909V (119905 119909)

119909119863119909119909V (119905 119909)

minus1

119905119910

119905

1198923119896lowast

(119905 119909) =

119896lowast if

119896

sum

119894=1

119894lowast

isin [0 1]

119896lowast

sum119896

119895=1119895lowast

if119896

sum

119894=1

119894lowast

gt 1


1198673(119905 119909

lowast) = minus

[119863119909V (119905 119909)]

2

2119863119909119909V (119905 119909)

(119910

119905)119879

minus1

119905119910

119905

(42)





0 = 119863119905V (119905 119909)


1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2

+(119909)2(119879119905) ]119863

119909119909V (119905 119909)

+ [119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+(120590119894)2

+119903119910

119905

119879

] 119909

+ [1199061

119905+ 1199062minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) [1198872(1199062) + 1198873(119909)]

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050)) 1198871(119909)

(43)



lowast=

119869(119892lowast) = E[V(119905

0 1199090)]



119869 (119892) = E [int

1199051

1199050

exp (minus119903119891(119897 minus 1199050))

times [1

21198882((1199062)2

(119897)) +1

21198883(119909119905minus 119897119903)2

] 119889119897

+1

21198881(119909 (1199051) minus 119897119903)2 exp (minus119903

119891(1199051minus 1199050)) ]

(44)


1198871(119909) =

1

21198881(119909 minus 119897

119903)2


1198872(1199062) =

1

21198882(1199062)2


(45)

1198873(119909) =

1

21198883(119909 minus 119897

119903)2

1198883isin (0infin) (46)


minus119905= minus

(119902119905)2

1198882

+ 1198883+ 1199021199052 (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+ 119902119905[minus119903119891minus119903119910

119905

119879

minus1

119905119910

119905+ (120590119890)2

+ (120590119894)2

] 1199021199051= 1198881

(47)

minus 119903

119905= minus

1198883(119909119903

119905minus 119897119903)

119902119905

minus 119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

minus [1199061

119905minus 119903119890

119905minus (120590119890)2

] minus (119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

)

minus (120590119894)2

119909119903(1199051) = 119897119903

(48)

minus 119897119905= minus119903119891119897119905+ 1198883(119909119903

119905minus 119897119903)2

minus 119902119905(120590119890)2

(119909119903

119905minus 1)2

minus 119902119905(120590119894)2

(119909119903

119905)2

119897 (1199051) = 0

(49)





1199062lowast

119905= minus

(119909 minus 119909119903

119905) 119902119905

1198882

1198922lowast

(119905 119909) = 1199062lowast

1198922lowast


lowast

119905= minus

(119909 minus 119909119903

119905)

119909minus1

119905119910

119905 1198923lowast

119879 timesX 997888rarr R119896

(50)

1198923119896lowast




(51)


V (119905 119909) = exp (minus119903119891(1199051minus 1199050)) [

1

2(119909 minus 119909

119903

119905)2

119902119905+

1

2119897119905] (52)

119869lowast= 119869 (119892

lowast) = E [V (119905

0 1199090)] (53)




minus119905= minus119887(119902

119905)2

+ 119888 + 2119886119905119902119905 1199021199051= 1198881

119887 =1

1198882 119888 = 119888

3isin (0infin) 119886 119879 997888rarr R

(54)


1199021

1199051= 119902 (119905

1minus 119905) 119886

1

119905= 119886 (119905

1minus 119905) 119902

1 [0 1199051minus 1199050] 997888rarr R

1

119905= minus (119905

1minus 119905) = minus119887(119902

1

119905)2

+ 119888 + 21198861

1199051199021

119905 1199021

1199050= 1198881

0 = 1

119905+ 119887(1199021

119905)2

minus 119888 minus 21198861

119905119902119905 1199021

0= 1198881

(55)


minus2

119905= minus119887(119902

2

119905)2

+ 119888 1199022

1199051= 1198881 119887 119888 isin (0infin) (56)


119905gt 0 for all 119905 isin 119879 = [0 119905

1minus 1199050]


1minus 1199050] 1199021119905

ge

1199022

119905gt 0 if


0 119887) ge (


119905

minus1198861

119905119887


(57)



(119886119905)2


1

1198882minus

1

1198882)

(58)


=


1minus 1199050]




1

1198882minus

1

1198882ge 0

(119886119905)2

le 1198884le (119888 minus 119888) (119887 minus 119887) = (119888

3minus 1198883) (

1

1198882minus

1

1198882)

(59)


1minus 1199050] 119902(1199051minus

119905) = 1199021

119905ge 1199022

119905gt 0




Note first that

V (119905 119909) = exp (minus119903119891(119905 minus 1199050)) [

1

2(119909 minus 119909

119903

119905)2

119902119905+

1

2119897119905]

119863119905V (119905 119909) = minus119903

119891V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))


119905) 119903

119905119902119905+

1

2(119909 minus 119909

119903

119905)2

119905+

1

2

119897119905]

119863119909V (119905 119909) = exp (minus119903

119891(119905 minus 1199050)) (119909 minus 119909

119903

119905) 119902119905

119863119909119909V (119905 119909) = exp (minus119903

119891(119905 minus 1199050)) 119902119905

(60)


So

0 = 119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))11986311990621198872(1199062)


119903

119905) 119902119905+ 11988821199062]

1199062lowast

= minus(119909 minus 119909

119903

119905) 119902119905

1198882

lowast= minus

(119909 minus 119909119903

119905) 119902119905

119909119902119905

minus1

119905119910

119905

(61)


1199021199051= 1198881 119909

119903(1199051) = 119897119903 119897 (119905

1) = 0

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050))

times [1

2(119909 minus 119909

119903(1199051))2

1199021199051+

1

2119897 (1199051)]

= exp (minus119903119891(1199051minus 1199050))

1

21198881(119909 minus 119897

119903)2

= exp (minus119903119891(1199051minus 1199050)) 1198871(119909)


exp (+119903119891(119905 minus 1199050))

times [119863119905V (119905 119909) +

1

2[(120590119890)2

(1 minus 119909)2

+(120590119894)2

(119909)2]119863119909119909V (119905 119909)

+ 119909 [119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]119863119909V (119905 119909)

+ [1199061

119905minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ 1199062lowast

119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050)) 1198872(1199062lowast

)

+ exp (minus119903119891(119905 minus 1199050)) 1198873(119909)

minus1

2

(119863119909V (119905 119909))

2

119863119909119909V (119905 119909)

119903119910

119905

119879

minus1

119905119910

119905]

= minus119903119891 1

2(119909 minus 119909

119903

119905)2

119902119905minus

1

2119903119891119897119905minus (119909 minus 119909

119903

119905) 119902119905119903

119905

+1

2(119909 minus 119909

119903

119905)2

119905+

1

2

119897119905

+1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2] 119902119905

+ (119909 minus 119909119903

119905) 119902119905[119909 (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+1199061

119905minus 119903119890

119905minus (120590119890)2

]

minus1

2

(119909 minus 119909119903

119905)2

(119902119905)2

1198882

+1

21198883(119909 minus 119897

119903)2

minus1

2(119909 minus 119909

119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

=1

2(119909)2[ minus 119903119891119902119905+ 119905+ (120590119890)2

119902119905+ (120590119894)2

119902119905

+ 2119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

minus(119902119905)2

1198882

+ 1198883minus 119902119905

119903119910

119905

119879

minus1

119905119910

119905]

+ 119909 [ + 119903119891119909119903

119905119902119905minus 119902119905119903

119905minus 119909119903

119905119905minus (120590119890)2

119902119905

minus 119909119903

119905119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ 119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] +119909119903

119905(119902119905)2

1198882

minus1198883119897119903+ 119909119903

119905119902119905

119903119910

119905

119879

minus1

119905119910

119905]

+1

2(119909)0[ minus 119903119891(119909119903

119905)2

119902119905+ 2119909119903

119905119902119905119903

119905+ (119909119903

119905)2

119905minus 119903119891119897119905

+ (120590119890)2

119902119905minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

]

minus(119909119903

119905)2

(119902119905)2

1198882

+ 1198883(119897119903)2

minus(119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905+ 119897119905]

(62)



minus 119902119905119903

119905minus 119909119903

119905119905+ 119903119891119909119903

119905119902119905minus 2(120590119890)2

119902119905

minus 119909119903

119905119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ 119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

]

+119909119903

119905(119902119905)2

1198882

minus 1198883119897119903+ 119909119903

119905119902119905

119903119910

119905

119879

minus1

119905119910

119905

= minus119902119905119903

119905

+ 119909119903

119905[minus

(119902119905)2

1198882

+ 1198883

+ 2119902119905(119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+119902119905((120590119890)2

+ (120590119894)2

minus 119903119891minus119903119910

119905

119879

minus1

119905119910

119905)]

minus 119909119903

119905119902119905[ (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

minus119903119891minus119903119910

119905

119879

minus1

119905119910

119905]

+ 119902 [minus(120590119890)2

+ (1199061

119905minus 119903119890

119905minus (120590119890)2

)] +119909119903

119905(119902119905)2

1198882

minus 1198883119897119903

= 119902119905[minus119903

119905+

1198883(119909119903

119905minus 119897119903)

119902119905

]

+ 119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ (119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

) + (120590119894)2

+ (1199061

119905minus 119903119890

119905minus (120590119890)2

)

= 0

(63)



119897119905minus 119903119891119897119905minus 119903119891(119909119903

119905)2

119902119905

+ 2119909119903

119905[119902119905119903

119905+ 119909119903

119905119905] minus (119909

119903

119905)2

119905+ (120590119890)2

119902119905

minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] + 1198883(119897119903)2

minus(119909119903

119905)2

(119902119905)2

1198882

minus (119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905


119897119905minus 119903119891119897119905minus 119903119891(119909119903

119905)2

119902119905+ (120590119890)2

119902119905

minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] + 1198883(119897119903)2

minus(119909119903

119905)2

(119902119905)2

1198882

minus (119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

+ 2119903119891(119909119903

119905)2

119902119905minus 2119909119903

119905119902119905(120590119890)2

minus 2(119909119903

119905)2

119902119905[119903

T+ 119903119890minus 119903119894+ (120590119890)2

+ (120590119894)2

]

+ 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] minus 11988831198971199032119909119903

119905

+2(119909119903

119905)2

(119902119905)2

1198882

+ 2(119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

minus(119909119903

119905)2

(119902119905)2

1198882

+ 1198883(119909119903

119905)2

+ (119909119903

119905)2

1199021199052 (119903

T+ 119903119890minus 119903119894+ (120590119890)2

+ (120590119894)2

)

+ (119909119903

119905)2

119902119905(minus119903119891+ (120590119890)2

+ (120590119894)2

minus119903119910

119905

119879

minus1

119905119910

119905)

= 119897119905minus 119903119891119897119905+ 1198883(119909119903

119905minus 119897119903)2

minus (120590119890)2

119902119905(119909119903

119905minus 1)2

minus (120590119894)2

119902119905(119909119903

119905)2

= 0

(64)







2gt 0 should


2(1199062) = (12)119888

2(1199062)2 given by (45) This







3(119909) = (12)119888



1198873(119909) = 119888

3[exp (119909 minus 119897

119903) + (119897119903minus 119909) minus 1] (65)







1199051198882 which





119903(119909) = (12)119888

3(119909 minus 119909




119903



119905119910



minus1

119905119910


deviation (119909 minus 119909119903

119905)




1199061

119905minus 119903119890

119905minus (120590119890)2

(66)



119890

119905is the


119903119890



1199051 119909119905lt 119897119903The term 119888

3(119909119903





119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

] (67)




T(119905) + (120590

119890)2+ (120590119894)2 is the effect of


(119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

) minus (120590119894)2

(68)




minus119903

119905= minus119896 (119909

119903

119905minus 119898) 119909

119903(1199051) = 119897119903

119896 = (119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2

+ (120590119894)2

)) +1198883

1199020

119898 =


) + (120590119890)2

(119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2+ (120590119894)2

)) + 11988831199020

(69)


119902119905

= 1199020 and lim

119905darr0119909119903




1199062lowast

119905= minus

(119909119905minus 119898) 119902

0

1198882

= gt 0 if 119909

119905lt 119898

lt 0 if 119909119905gt 119898

120587lowast

119905= minus

(119909119905minus 119898)

119909119905

119895

= gt 0 if 119909

119905lt 119898


(70)


Case 1 (119909119905











119898 =


) + (120590119890)2

(119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2+ (120590119894)2

)) + 11988831199020

(71)





Appendices













References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014






0 = 119863119905V (119905 119909)


1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2

+(119909)2(119879119905) ]119863

119909119909V (119905 119909)

+ [119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+(120590119894)2

+119903119910

119905

119879

] 119909

+ [1199061

119905+ 1199062minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ exp (minus119903119891(119905 minus 1199050)) [1198872(1199062) + 1198873(119909)]

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050)) 1198871(119909)

(43)



lowast=

119869(119892lowast) = E[V(119905

0 1199090)]



119869 (119892) = E [int

1199051

1199050

exp (minus119903119891(119897 minus 1199050))

times [1

21198882((1199062)2

(119897)) +1

21198883(119909119905minus 119897119903)2

] 119889119897

+1

21198881(119909 (1199051) minus 119897119903)2 exp (minus119903

119891(1199051minus 1199050)) ]

(44)


1198871(119909) =

1

21198881(119909 minus 119897

119903)2


1198872(1199062) =

1

21198882(1199062)2


(45)

1198873(119909) =

1

21198883(119909 minus 119897

119903)2

1198883isin (0infin) (46)


minus119905= minus

(119902119905)2

1198882

+ 1198883+ 1199021199052 (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+ 119902119905[minus119903119891minus119903119910

119905

119879

minus1

119905119910

119905+ (120590119890)2

+ (120590119894)2

] 1199021199051= 1198881

(47)

minus 119903

119905= minus

1198883(119909119903

119905minus 119897119903)

119902119905

minus 119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

minus [1199061

119905minus 119903119890

119905minus (120590119890)2

] minus (119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

)

minus (120590119894)2

119909119903(1199051) = 119897119903

(48)

minus 119897119905= minus119903119891119897119905+ 1198883(119909119903

119905minus 119897119903)2

minus 119902119905(120590119890)2

(119909119903

119905minus 1)2

minus 119902119905(120590119894)2

(119909119903

119905)2

119897 (1199051) = 0

(49)





1199062lowast

119905= minus

(119909 minus 119909119903

119905) 119902119905

1198882

1198922lowast

(119905 119909) = 1199062lowast

1198922lowast


lowast

119905= minus

(119909 minus 119909119903

119905)

119909minus1

119905119910

119905 1198923lowast

119879 timesX 997888rarr R119896

(50)

1198923119896lowast




(51)


V (119905 119909) = exp (minus119903119891(1199051minus 1199050)) [

1

2(119909 minus 119909

119903

119905)2

119902119905+

1

2119897119905] (52)

119869lowast= 119869 (119892

lowast) = E [V (119905

0 1199090)] (53)




minus119905= minus119887(119902

119905)2

+ 119888 + 2119886119905119902119905 1199021199051= 1198881

119887 =1

1198882 119888 = 119888

3isin (0infin) 119886 119879 997888rarr R

(54)


1199021

1199051= 119902 (119905

1minus 119905) 119886

1

119905= 119886 (119905

1minus 119905) 119902

1 [0 1199051minus 1199050] 997888rarr R

1

119905= minus (119905

1minus 119905) = minus119887(119902

1

119905)2

+ 119888 + 21198861

1199051199021

119905 1199021

1199050= 1198881

0 = 1

119905+ 119887(1199021

119905)2

minus 119888 minus 21198861

119905119902119905 1199021

0= 1198881

(55)


minus2

119905= minus119887(119902

2

119905)2

+ 119888 1199022

1199051= 1198881 119887 119888 isin (0infin) (56)


119905gt 0 for all 119905 isin 119879 = [0 119905

1minus 1199050]


1minus 1199050] 1199021119905

ge

1199022

119905gt 0 if


0 119887) ge (


119905

minus1198861

119905119887


(57)



(119886119905)2


1

1198882minus

1

1198882)

(58)


=


1minus 1199050]




1

1198882minus

1

1198882ge 0

(119886119905)2

le 1198884le (119888 minus 119888) (119887 minus 119887) = (119888

3minus 1198883) (

1

1198882minus

1

1198882)

(59)


1minus 1199050] 119902(1199051minus

119905) = 1199021

119905ge 1199022

119905gt 0




Note first that

V (119905 119909) = exp (minus119903119891(119905 minus 1199050)) [

1

2(119909 minus 119909

119903

119905)2

119902119905+

1

2119897119905]

119863119905V (119905 119909) = minus119903

119891V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))


119905) 119903

119905119902119905+

1

2(119909 minus 119909

119903

119905)2

119905+

1

2

119897119905]

119863119909V (119905 119909) = exp (minus119903

119891(119905 minus 1199050)) (119909 minus 119909

119903

119905) 119902119905

119863119909119909V (119905 119909) = exp (minus119903

119891(119905 minus 1199050)) 119902119905

(60)


So

0 = 119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))11986311990621198872(1199062)


119903

119905) 119902119905+ 11988821199062]

1199062lowast

= minus(119909 minus 119909

119903

119905) 119902119905

1198882

lowast= minus

(119909 minus 119909119903

119905) 119902119905

119909119902119905

minus1

119905119910

119905

(61)


1199021199051= 1198881 119909

119903(1199051) = 119897119903 119897 (119905

1) = 0

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050))

times [1

2(119909 minus 119909

119903(1199051))2

1199021199051+

1

2119897 (1199051)]

= exp (minus119903119891(1199051minus 1199050))

1

21198881(119909 minus 119897

119903)2

= exp (minus119903119891(1199051minus 1199050)) 1198871(119909)


exp (+119903119891(119905 minus 1199050))

times [119863119905V (119905 119909) +

1

2[(120590119890)2

(1 minus 119909)2

+(120590119894)2

(119909)2]119863119909119909V (119905 119909)

+ 119909 [119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]119863119909V (119905 119909)

+ [1199061

119905minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ 1199062lowast

119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050)) 1198872(1199062lowast

)

+ exp (minus119903119891(119905 minus 1199050)) 1198873(119909)

minus1

2

(119863119909V (119905 119909))

2

119863119909119909V (119905 119909)

119903119910

119905

119879

minus1

119905119910

119905]

= minus119903119891 1

2(119909 minus 119909

119903

119905)2

119902119905minus

1

2119903119891119897119905minus (119909 minus 119909

119903

119905) 119902119905119903

119905

+1

2(119909 minus 119909

119903

119905)2

119905+

1

2

119897119905

+1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2] 119902119905

+ (119909 minus 119909119903

119905) 119902119905[119909 (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+1199061

119905minus 119903119890

119905minus (120590119890)2

]

minus1

2

(119909 minus 119909119903

119905)2

(119902119905)2

1198882

+1

21198883(119909 minus 119897

119903)2

minus1

2(119909 minus 119909

119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

=1

2(119909)2[ minus 119903119891119902119905+ 119905+ (120590119890)2

119902119905+ (120590119894)2

119902119905

+ 2119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

minus(119902119905)2

1198882

+ 1198883minus 119902119905

119903119910

119905

119879

minus1

119905119910

119905]

+ 119909 [ + 119903119891119909119903

119905119902119905minus 119902119905119903

119905minus 119909119903

119905119905minus (120590119890)2

119902119905

minus 119909119903

119905119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ 119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] +119909119903

119905(119902119905)2

1198882

minus1198883119897119903+ 119909119903

119905119902119905

119903119910

119905

119879

minus1

119905119910

119905]

+1

2(119909)0[ minus 119903119891(119909119903

119905)2

119902119905+ 2119909119903

119905119902119905119903

119905+ (119909119903

119905)2

119905minus 119903119891119897119905

+ (120590119890)2

119902119905minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

]

minus(119909119903

119905)2

(119902119905)2

1198882

+ 1198883(119897119903)2

minus(119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905+ 119897119905]

(62)



minus 119902119905119903

119905minus 119909119903

119905119905+ 119903119891119909119903

119905119902119905minus 2(120590119890)2

119902119905

minus 119909119903

119905119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ 119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

]

+119909119903

119905(119902119905)2

1198882

minus 1198883119897119903+ 119909119903

119905119902119905

119903119910

119905

119879

minus1

119905119910

119905

= minus119902119905119903

119905

+ 119909119903

119905[minus

(119902119905)2

1198882

+ 1198883

+ 2119902119905(119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+119902119905((120590119890)2

+ (120590119894)2

minus 119903119891minus119903119910

119905

119879

minus1

119905119910

119905)]

minus 119909119903

119905119902119905[ (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

minus119903119891minus119903119910

119905

119879

minus1

119905119910

119905]

+ 119902 [minus(120590119890)2

+ (1199061

119905minus 119903119890

119905minus (120590119890)2

)] +119909119903

119905(119902119905)2

1198882

minus 1198883119897119903

= 119902119905[minus119903

119905+

1198883(119909119903

119905minus 119897119903)

119902119905

]

+ 119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ (119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

) + (120590119894)2

+ (1199061

119905minus 119903119890

119905minus (120590119890)2

)

= 0

(63)



119897119905minus 119903119891119897119905minus 119903119891(119909119903

119905)2

119902119905

+ 2119909119903

119905[119902119905119903

119905+ 119909119903

119905119905] minus (119909

119903

119905)2

119905+ (120590119890)2

119902119905

minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] + 1198883(119897119903)2

minus(119909119903

119905)2

(119902119905)2

1198882

minus (119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905


119897119905minus 119903119891119897119905minus 119903119891(119909119903

119905)2

119902119905+ (120590119890)2

119902119905

minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] + 1198883(119897119903)2

minus(119909119903

119905)2

(119902119905)2

1198882

minus (119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

+ 2119903119891(119909119903

119905)2

119902119905minus 2119909119903

119905119902119905(120590119890)2

minus 2(119909119903

119905)2

119902119905[119903

T+ 119903119890minus 119903119894+ (120590119890)2

+ (120590119894)2

]

+ 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] minus 11988831198971199032119909119903

119905

+2(119909119903

119905)2

(119902119905)2

1198882

+ 2(119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

minus(119909119903

119905)2

(119902119905)2

1198882

+ 1198883(119909119903

119905)2

+ (119909119903

119905)2

1199021199052 (119903

T+ 119903119890minus 119903119894+ (120590119890)2

+ (120590119894)2

)

+ (119909119903

119905)2

119902119905(minus119903119891+ (120590119890)2

+ (120590119894)2

minus119903119910

119905

119879

minus1

119905119910

119905)

= 119897119905minus 119903119891119897119905+ 1198883(119909119903

119905minus 119897119903)2

minus (120590119890)2

119902119905(119909119903

119905minus 1)2

minus (120590119894)2

119902119905(119909119903

119905)2

= 0

(64)







2gt 0 should


2(1199062) = (12)119888

2(1199062)2 given by (45) This







3(119909) = (12)119888



1198873(119909) = 119888

3[exp (119909 minus 119897

119903) + (119897119903minus 119909) minus 1] (65)







1199051198882 which





119903(119909) = (12)119888

3(119909 minus 119909




119903



119905119910



minus1

119905119910


deviation (119909 minus 119909119903

119905)




1199061

119905minus 119903119890

119905minus (120590119890)2

(66)



119890

119905is the


119903119890



1199051 119909119905lt 119897119903The term 119888

3(119909119903





119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

] (67)




T(119905) + (120590

119890)2+ (120590119894)2 is the effect of


(119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

) minus (120590119894)2

(68)




minus119903

119905= minus119896 (119909

119903

119905minus 119898) 119909

119903(1199051) = 119897119903

119896 = (119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2

+ (120590119894)2

)) +1198883

1199020

119898 =


) + (120590119890)2

(119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2+ (120590119894)2

)) + 11988831199020

(69)


119902119905

= 1199020 and lim

119905darr0119909119903




1199062lowast

119905= minus

(119909119905minus 119898) 119902

0

1198882

= gt 0 if 119909

119905lt 119898

lt 0 if 119909119905gt 119898

120587lowast

119905= minus

(119909119905minus 119898)

119909119905

119895

= gt 0 if 119909

119905lt 119898


(70)


Case 1 (119909119905











119898 =


) + (120590119890)2

(119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2+ (120590119894)2

)) + 11988831199020

(71)





Appendices













References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014






minus119905= minus119887(119902

119905)2

+ 119888 + 2119886119905119902119905 1199021199051= 1198881

119887 =1

1198882 119888 = 119888

3isin (0infin) 119886 119879 997888rarr R

(54)


1199021

1199051= 119902 (119905

1minus 119905) 119886

1

119905= 119886 (119905

1minus 119905) 119902

1 [0 1199051minus 1199050] 997888rarr R

1

119905= minus (119905

1minus 119905) = minus119887(119902

1

119905)2

+ 119888 + 21198861

1199051199021

119905 1199021

1199050= 1198881

0 = 1

119905+ 119887(1199021

119905)2

minus 119888 minus 21198861

119905119902119905 1199021

0= 1198881

(55)


minus2

119905= minus119887(119902

2

119905)2

+ 119888 1199022

1199051= 1198881 119887 119888 isin (0infin) (56)


119905gt 0 for all 119905 isin 119879 = [0 119905

1minus 1199050]


1minus 1199050] 1199021119905

ge

1199022

119905gt 0 if


0 119887) ge (


119905

minus1198861

119905119887


(57)



(119886119905)2


1

1198882minus

1

1198882)

(58)


=


1minus 1199050]




1

1198882minus

1

1198882ge 0

(119886119905)2

le 1198884le (119888 minus 119888) (119887 minus 119887) = (119888

3minus 1198883) (

1

1198882minus

1

1198882)

(59)


1minus 1199050] 119902(1199051minus

119905) = 1199021

119905ge 1199022

119905gt 0




Note first that

V (119905 119909) = exp (minus119903119891(119905 minus 1199050)) [

1

2(119909 minus 119909

119903

119905)2

119902119905+

1

2119897119905]

119863119905V (119905 119909) = minus119903

119891V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))


119905) 119903

119905119902119905+

1

2(119909 minus 119909

119903

119905)2

119905+

1

2

119897119905]

119863119909V (119905 119909) = exp (minus119903

119891(119905 minus 1199050)) (119909 minus 119909

119903

119905) 119902119905

119863119909119909V (119905 119909) = exp (minus119903

119891(119905 minus 1199050)) 119902119905

(60)


So

0 = 119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050))11986311990621198872(1199062)


119903

119905) 119902119905+ 11988821199062]

1199062lowast

= minus(119909 minus 119909

119903

119905) 119902119905

1198882

lowast= minus

(119909 minus 119909119903

119905) 119902119905

119909119902119905

minus1

119905119910

119905

(61)


1199021199051= 1198881 119909

119903(1199051) = 119897119903 119897 (119905

1) = 0

V (1199051 119909) = exp (minus119903

119891(1199051minus 1199050))

times [1

2(119909 minus 119909

119903(1199051))2

1199021199051+

1

2119897 (1199051)]

= exp (minus119903119891(1199051minus 1199050))

1

21198881(119909 minus 119897

119903)2

= exp (minus119903119891(1199051minus 1199050)) 1198871(119909)


exp (+119903119891(119905 minus 1199050))

times [119863119905V (119905 119909) +

1

2[(120590119890)2

(1 minus 119909)2

+(120590119894)2

(119909)2]119863119909119909V (119905 119909)

+ 119909 [119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]119863119909V (119905 119909)

+ [1199061

119905minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ 1199062lowast

119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050)) 1198872(1199062lowast

)

+ exp (minus119903119891(119905 minus 1199050)) 1198873(119909)

minus1

2

(119863119909V (119905 119909))

2

119863119909119909V (119905 119909)

119903119910

119905

119879

minus1

119905119910

119905]

= minus119903119891 1

2(119909 minus 119909

119903

119905)2

119902119905minus

1

2119903119891119897119905minus (119909 minus 119909

119903

119905) 119902119905119903

119905

+1

2(119909 minus 119909

119903

119905)2

119905+

1

2

119897119905

+1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2] 119902119905

+ (119909 minus 119909119903

119905) 119902119905[119909 (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+1199061

119905minus 119903119890

119905minus (120590119890)2

]

minus1

2

(119909 minus 119909119903

119905)2

(119902119905)2

1198882

+1

21198883(119909 minus 119897

119903)2

minus1

2(119909 minus 119909

119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

=1

2(119909)2[ minus 119903119891119902119905+ 119905+ (120590119890)2

119902119905+ (120590119894)2

119902119905

+ 2119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

minus(119902119905)2

1198882

+ 1198883minus 119902119905

119903119910

119905

119879

minus1

119905119910

119905]

+ 119909 [ + 119903119891119909119903

119905119902119905minus 119902119905119903

119905minus 119909119903

119905119905minus (120590119890)2

119902119905

minus 119909119903

119905119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ 119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] +119909119903

119905(119902119905)2

1198882

minus1198883119897119903+ 119909119903

119905119902119905

119903119910

119905

119879

minus1

119905119910

119905]

+1

2(119909)0[ minus 119903119891(119909119903

119905)2

119902119905+ 2119909119903

119905119902119905119903

119905+ (119909119903

119905)2

119905minus 119903119891119897119905

+ (120590119890)2

119902119905minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

]

minus(119909119903

119905)2

(119902119905)2

1198882

+ 1198883(119897119903)2

minus(119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905+ 119897119905]

(62)



minus 119902119905119903

119905minus 119909119903

119905119905+ 119903119891119909119903

119905119902119905minus 2(120590119890)2

119902119905

minus 119909119903

119905119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ 119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

]

+119909119903

119905(119902119905)2

1198882

minus 1198883119897119903+ 119909119903

119905119902119905

119903119910

119905

119879

minus1

119905119910

119905

= minus119902119905119903

119905

+ 119909119903

119905[minus

(119902119905)2

1198882

+ 1198883

+ 2119902119905(119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+119902119905((120590119890)2

+ (120590119894)2

minus 119903119891minus119903119910

119905

119879

minus1

119905119910

119905)]

minus 119909119903

119905119902119905[ (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

minus119903119891minus119903119910

119905

119879

minus1

119905119910

119905]

+ 119902 [minus(120590119890)2

+ (1199061

119905minus 119903119890

119905minus (120590119890)2

)] +119909119903

119905(119902119905)2

1198882

minus 1198883119897119903

= 119902119905[minus119903

119905+

1198883(119909119903

119905minus 119897119903)

119902119905

]

+ 119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ (119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

) + (120590119894)2

+ (1199061

119905minus 119903119890

119905minus (120590119890)2

)

= 0

(63)



119897119905minus 119903119891119897119905minus 119903119891(119909119903

119905)2

119902119905

+ 2119909119903

119905[119902119905119903

119905+ 119909119903

119905119905] minus (119909

119903

119905)2

119905+ (120590119890)2

119902119905

minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] + 1198883(119897119903)2

minus(119909119903

119905)2

(119902119905)2

1198882

minus (119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905


119897119905minus 119903119891119897119905minus 119903119891(119909119903

119905)2

119902119905+ (120590119890)2

119902119905

minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] + 1198883(119897119903)2

minus(119909119903

119905)2

(119902119905)2

1198882

minus (119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

+ 2119903119891(119909119903

119905)2

119902119905minus 2119909119903

119905119902119905(120590119890)2

minus 2(119909119903

119905)2

119902119905[119903

T+ 119903119890minus 119903119894+ (120590119890)2

+ (120590119894)2

]

+ 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] minus 11988831198971199032119909119903

119905

+2(119909119903

119905)2

(119902119905)2

1198882

+ 2(119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

minus(119909119903

119905)2

(119902119905)2

1198882

+ 1198883(119909119903

119905)2

+ (119909119903

119905)2

1199021199052 (119903

T+ 119903119890minus 119903119894+ (120590119890)2

+ (120590119894)2

)

+ (119909119903

119905)2

119902119905(minus119903119891+ (120590119890)2

+ (120590119894)2

minus119903119910

119905

119879

minus1

119905119910

119905)

= 119897119905minus 119903119891119897119905+ 1198883(119909119903

119905minus 119897119903)2

minus (120590119890)2

119902119905(119909119903

119905minus 1)2

minus (120590119894)2

119902119905(119909119903

119905)2

= 0

(64)







2gt 0 should


2(1199062) = (12)119888

2(1199062)2 given by (45) This







3(119909) = (12)119888



1198873(119909) = 119888

3[exp (119909 minus 119897

119903) + (119897119903minus 119909) minus 1] (65)







1199051198882 which





119903(119909) = (12)119888

3(119909 minus 119909




119903



119905119910



minus1

119905119910


deviation (119909 minus 119909119903

119905)




1199061

119905minus 119903119890

119905minus (120590119890)2

(66)



119890

119905is the


119903119890



1199051 119909119905lt 119897119903The term 119888

3(119909119903





119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

] (67)




T(119905) + (120590

119890)2+ (120590119894)2 is the effect of


(119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

) minus (120590119894)2

(68)




minus119903

119905= minus119896 (119909

119903

119905minus 119898) 119909

119903(1199051) = 119897119903

119896 = (119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2

+ (120590119894)2

)) +1198883

1199020

119898 =


) + (120590119890)2

(119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2+ (120590119894)2

)) + 11988831199020

(69)


119902119905

= 1199020 and lim

119905darr0119909119903




1199062lowast

119905= minus

(119909119905minus 119898) 119902

0

1198882

= gt 0 if 119909

119905lt 119898

lt 0 if 119909119905gt 119898

120587lowast

119905= minus

(119909119905minus 119898)

119909119905

119895

= gt 0 if 119909

119905lt 119898


(70)


Case 1 (119909119905











119898 =


) + (120590119890)2

(119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2+ (120590119894)2

)) + 11988831199020

(71)





Appendices













References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




exp (+119903119891(119905 minus 1199050))

times [119863119905V (119905 119909) +

1

2[(120590119890)2

(1 minus 119909)2

+(120590119894)2

(119909)2]119863119909119909V (119905 119909)

+ 119909 [119903T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]119863119909V (119905 119909)

+ [1199061

119905minus 119903119890

119905minus (120590119890)2

]119863119909V (119905 119909)

+ 1199062lowast

119863119909V (119905 119909) + exp (minus119903

119891(119905 minus 1199050)) 1198872(1199062lowast

)

+ exp (minus119903119891(119905 minus 1199050)) 1198873(119909)

minus1

2

(119863119909V (119905 119909))

2

119863119909119909V (119905 119909)

119903119910

119905

119879

minus1

119905119910

119905]

= minus119903119891 1

2(119909 minus 119909

119903

119905)2

119902119905minus

1

2119903119891119897119905minus (119909 minus 119909

119903

119905) 119902119905119903

119905

+1

2(119909 minus 119909

119903

119905)2

119905+

1

2

119897119905

+1

2[(120590119890)2

(1 minus 119909)2+ (120590119894)2

(119909)2] 119902119905

+ (119909 minus 119909119903

119905) 119902119905[119909 (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+1199061

119905minus 119903119890

119905minus (120590119890)2

]

minus1

2

(119909 minus 119909119903

119905)2

(119902119905)2

1198882

+1

21198883(119909 minus 119897

119903)2

minus1

2(119909 minus 119909

119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

=1

2(119909)2[ minus 119903119891119902119905+ 119905+ (120590119890)2

119902119905+ (120590119894)2

119902119905

+ 2119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

minus(119902119905)2

1198882

+ 1198883minus 119902119905

119903119910

119905

119879

minus1

119905119910

119905]

+ 119909 [ + 119903119891119909119903

119905119902119905minus 119902119905119903

119905minus 119909119903

119905119905minus (120590119890)2

119902119905

minus 119909119903

119905119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ 119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] +119909119903

119905(119902119905)2

1198882

minus1198883119897119903+ 119909119903

119905119902119905

119903119910

119905

119879

minus1

119905119910

119905]

+1

2(119909)0[ minus 119903119891(119909119903

119905)2

119902119905+ 2119909119903

119905119902119905119903

119905+ (119909119903

119905)2

119905minus 119903119891119897119905

+ (120590119890)2

119902119905minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

]

minus(119909119903

119905)2

(119902119905)2

1198882

+ 1198883(119897119903)2

minus(119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905+ 119897119905]

(62)



minus 119902119905119903

119905minus 119909119903

119905119905+ 119903119891119909119903

119905119902119905minus 2(120590119890)2

119902119905

minus 119909119903

119905119902119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ 119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

]

+119909119903

119905(119902119905)2

1198882

minus 1198883119897119903+ 119909119903

119905119902119905

119903119910

119905

119879

minus1

119905119910

119905

= minus119902119905119903

119905

+ 119909119903

119905[minus

(119902119905)2

1198882

+ 1198883

+ 2119902119905(119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

+119902119905((120590119890)2

+ (120590119894)2

minus 119903119891minus119903119910

119905

119879

minus1

119905119910

119905)]

minus 119909119903

119905119902119905[ (119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

)

minus119903119891minus119903119910

119905

119879

minus1

119905119910

119905]

+ 119902 [minus(120590119890)2

+ (1199061

119905minus 119903119890

119905minus (120590119890)2

)] +119909119903

119905(119902119905)2

1198882

minus 1198883119897119903

= 119902119905[minus119903

119905+

1198883(119909119903

119905minus 119897119903)

119902119905

]

+ 119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

]

+ (119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

) + (120590119894)2

+ (1199061

119905minus 119903119890

119905minus (120590119890)2

)

= 0

(63)



119897119905minus 119903119891119897119905minus 119903119891(119909119903

119905)2

119902119905

+ 2119909119903

119905[119902119905119903

119905+ 119909119903

119905119905] minus (119909

119903

119905)2

119905+ (120590119890)2

119902119905

minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] + 1198883(119897119903)2

minus(119909119903

119905)2

(119902119905)2

1198882

minus (119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905


119897119905minus 119903119891119897119905minus 119903119891(119909119903

119905)2

119902119905+ (120590119890)2

119902119905

minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] + 1198883(119897119903)2

minus(119909119903

119905)2

(119902119905)2

1198882

minus (119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

+ 2119903119891(119909119903

119905)2

119902119905minus 2119909119903

119905119902119905(120590119890)2

minus 2(119909119903

119905)2

119902119905[119903

T+ 119903119890minus 119903119894+ (120590119890)2

+ (120590119894)2

]

+ 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] minus 11988831198971199032119909119903

119905

+2(119909119903

119905)2

(119902119905)2

1198882

+ 2(119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

minus(119909119903

119905)2

(119902119905)2

1198882

+ 1198883(119909119903

119905)2

+ (119909119903

119905)2

1199021199052 (119903

T+ 119903119890minus 119903119894+ (120590119890)2

+ (120590119894)2

)

+ (119909119903

119905)2

119902119905(minus119903119891+ (120590119890)2

+ (120590119894)2

minus119903119910

119905

119879

minus1

119905119910

119905)

= 119897119905minus 119903119891119897119905+ 1198883(119909119903

119905minus 119897119903)2

minus (120590119890)2

119902119905(119909119903

119905minus 1)2

minus (120590119894)2

119902119905(119909119903

119905)2

= 0

(64)







2gt 0 should


2(1199062) = (12)119888

2(1199062)2 given by (45) This







3(119909) = (12)119888



1198873(119909) = 119888

3[exp (119909 minus 119897

119903) + (119897119903minus 119909) minus 1] (65)







1199051198882 which





119903(119909) = (12)119888

3(119909 minus 119909




119903



119905119910



minus1

119905119910


deviation (119909 minus 119909119903

119905)




1199061

119905minus 119903119890

119905minus (120590119890)2

(66)



119890

119905is the


119903119890



1199051 119909119905lt 119897119903The term 119888

3(119909119903





119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

] (67)




T(119905) + (120590

119890)2+ (120590119894)2 is the effect of


(119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

) minus (120590119894)2

(68)




minus119903

119905= minus119896 (119909

119903

119905minus 119898) 119909

119903(1199051) = 119897119903

119896 = (119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2

+ (120590119894)2

)) +1198883

1199020

119898 =


) + (120590119890)2

(119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2+ (120590119894)2

)) + 11988831199020

(69)


119902119905

= 1199020 and lim

119905darr0119909119903




1199062lowast

119905= minus

(119909119905minus 119898) 119902

0

1198882

= gt 0 if 119909

119905lt 119898

lt 0 if 119909119905gt 119898

120587lowast

119905= minus

(119909119905minus 119898)

119909119905

119895

= gt 0 if 119909

119905lt 119898


(70)


Case 1 (119909119905











119898 =


) + (120590119890)2

(119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2+ (120590119894)2

)) + 11988831199020

(71)





Appendices













References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





119897119905minus 119903119891119897119905minus 119903119891(119909119903

119905)2

119902119905

+ 2119909119903

119905[119902119905119903

119905+ 119909119903

119905119905] minus (119909

119903

119905)2

119905+ (120590119890)2

119902119905

minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] + 1198883(119897119903)2

minus(119909119903

119905)2

(119902119905)2

1198882

minus (119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905


119897119905minus 119903119891119897119905minus 119903119891(119909119903

119905)2

119902119905+ (120590119890)2

119902119905

minus 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] + 1198883(119897119903)2

minus(119909119903

119905)2

(119902119905)2

1198882

minus (119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

+ 2119903119891(119909119903

119905)2

119902119905minus 2119909119903

119905119902119905(120590119890)2

minus 2(119909119903

119905)2

119902119905[119903

T+ 119903119890minus 119903119894+ (120590119890)2

+ (120590119894)2

]

+ 2119909119903

119905119902119905[1199061

119905minus 119903119890

119905minus (120590119890)2

] minus 11988831198971199032119909119903

119905

+2(119909119903

119905)2

(119902119905)2

1198882

+ 2(119909119903

119905)2

119902119905

119903119910

119905

119879

minus1

119905119910

119905

minus(119909119903

119905)2

(119902119905)2

1198882

+ 1198883(119909119903

119905)2

+ (119909119903

119905)2

1199021199052 (119903

T+ 119903119890minus 119903119894+ (120590119890)2

+ (120590119894)2

)

+ (119909119903

119905)2

119902119905(minus119903119891+ (120590119890)2

+ (120590119894)2

minus119903119910

119905

119879

minus1

119905119910

119905)

= 119897119905minus 119903119891119897119905+ 1198883(119909119903

119905minus 119897119903)2

minus (120590119890)2

119902119905(119909119903

119905minus 1)2

minus (120590119894)2

119902119905(119909119903

119905)2

= 0

(64)







2gt 0 should


2(1199062) = (12)119888

2(1199062)2 given by (45) This







3(119909) = (12)119888



1198873(119909) = 119888

3[exp (119909 minus 119897

119903) + (119897119903minus 119909) minus 1] (65)







1199051198882 which





119903(119909) = (12)119888

3(119909 minus 119909




119903



119905119910



minus1

119905119910


deviation (119909 minus 119909119903

119905)




1199061

119905minus 119903119890

119905minus (120590119890)2

(66)



119890

119905is the


119903119890



1199051 119909119905lt 119897119903The term 119888

3(119909119903





119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

] (67)




T(119905) + (120590

119890)2+ (120590119894)2 is the effect of


(119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

) minus (120590119894)2

(68)




minus119903

119905= minus119896 (119909

119903

119905minus 119898) 119909

119903(1199051) = 119897119903

119896 = (119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2

+ (120590119894)2

)) +1198883

1199020

119898 =


) + (120590119890)2

(119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2+ (120590119894)2

)) + 11988831199020

(69)


119902119905

= 1199020 and lim

119905darr0119909119903




1199062lowast

119905= minus

(119909119905minus 119898) 119902

0

1198882

= gt 0 if 119909

119905lt 119898

lt 0 if 119909119905gt 119898

120587lowast

119905= minus

(119909119905minus 119898)

119909119905

119895

= gt 0 if 119909

119905lt 119898


(70)


Case 1 (119909119905











119898 =


) + (120590119890)2

(119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2+ (120590119894)2

)) + 11988831199020

(71)





Appendices













References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





1199061

119905minus 119903119890

119905minus (120590119890)2

(66)



119890

119905is the


119903119890



1199051 119909119905lt 119897119903The term 119888

3(119909119903





119909119903

119905[119903

T(119905) + 119903

119890

119905minus 119903119894

119905+ (120590119890)2

+ (120590119894)2

] (67)




T(119905) + (120590

119890)2+ (120590119894)2 is the effect of


(119909119903

119905minus 1) ((120590

119890)2

+ (120590119894)2

) minus (120590119894)2

(68)




minus119903

119905= minus119896 (119909

119903

119905minus 119898) 119909

119903(1199051) = 119897119903

119896 = (119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2

+ (120590119894)2

)) +1198883

1199020

119898 =


) + (120590119890)2

(119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2+ (120590119894)2

)) + 11988831199020

(69)


119902119905

= 1199020 and lim

119905darr0119909119903




1199062lowast

119905= minus

(119909119905minus 119898) 119902

0

1198882

= gt 0 if 119909

119905lt 119898

lt 0 if 119909119905gt 119898

120587lowast

119905= minus

(119909119905minus 119898)

119909119905

119895

= gt 0 if 119909

119905lt 119898


(70)


Case 1 (119909119905











119898 =


) + (120590119890)2

(119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2+ (120590119894)2

)) + 11988831199020

(71)





Appendices













References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014








119898 =


) + (120590119890)2

(119903T+ 119903119890minus 119903119894+ 2 ((120590

119890)2+ (120590119894)2

)) + 11988831199020

(71)





Appendices













References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014



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