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Exact solutions and doubly efficient approximations ofjump-diffusion itô equationsYoosef Maghsoodi aa Department of Mathematics , University of Southampton , Southampton, SO17 1BJ, UK Fax:Published online: 03 Apr 2007.

To cite this article: Yoosef Maghsoodi (1998) Exact solutions and doubly efficient approximations of jump-diffusion itôequations, Stochastic Analysis and Applications, 16:6, 1049-1072, DOI: 10.1080/07362999808809579

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STOCHASTIC ANALYSIS AND APPLICATIONS, 16(6), 1049-1 072 (1 998)

EXACT SOLUTIONS AND DOUBLY EFFICIENT APPROXIMATIONS

OF JUMP-DIFFUSION ITO EQUATIONS

Yoosef Maghsoodi

Department of Mathematics, University of Southampton

Southampton SO17 lBJ , UK, Fax: +44 1703 342321

ABSTRACT

This paper presents exact solutions to an unsolved class of jump-diffusion stochastic differential equations and derives efficient numerical schemes for the general non-linear cases. It is proved that even the second order mean square efficient schemes may not be second order efficient in the weak sense. The generator and Taylor expansion in the expectations semi-group are verified under weaker conditions and applied to derive new doub ly efficient schemes which are proved to converge with t,he best possible order rate in both senses. A class of direct j u m p - a d a p t e d schemes are also presented. Comparative simulations are consistent with the findings.

1. INTRODUCTION

In modelling, filtering and control of continuous time stochastic systems often in addi-

tion to the continuously acting white noise type disturbances the effect of random impu!ses

at random times have to be represented too. Such systems have been modelled by Jump-

diffusion processes which are solutions of stochastic differential equations driven simulta-

neously by Brownian motion and the Poisson counting process. Let ( R , F , P) denote the

basic ~ r o b a b i l i t ~ space with the filteration {F,; t 6 [t,,T]} satisfying the usual properties.

The Rn valued adapted Markov process { x t ; t E [ to ,U} is said to be the solution of the

n-dimensional jump-diffusion stochastic differential equation if it satisfies:

where without loss of generality the initial state is assumed to be non-random, i.e. xto = xn

w.p.l., a and c are R" valued and u an n x q matrix valued continuously differentiable

function. The second integral in (1.1) is defined as the It8 integral with respect to the

q-dimensional standard Brownian motion process {W,; 3i,t E [ to ,q} and the third inte-

gral is with respect to the inhomogeneous Poisson counting process { N t ; 3i, t E [t,,T])}

Copyright Q 1998 by Marcel Dekker, Inc.

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1050 MAGHSOODI

which is independent of the Brownian motion and represents the random number of jumps

affecting the system before time t and has the Poisson distribution with parameter At which

is bounded and continuously differentiable. This integral is defined in terms of It6 integrals

with respect to the Poisson random measure (Ito [9], Gikhman and Skorokhod [5]). Exis-

tence and pathwise uniqueness of the solution follows under growth restriction, (uniform)

Lipschitz and continuity conditions on the functions a , a , and c (see [5],[9],[25]). Appendix

A contains some useful properties of the integrals and the solution of (1.1). When the

function c vanishes identically (1.1) will be referred to as SDE1. No closed form solutions

are available for the general non-linear case. This paper presents solutions to an unsolved

non-linear class and for the general non-linear cases, derives efficient difference formulae

which converge to the solution of (1.1) with the best possible order rate in the grid size,

both in the weak sense as well as in mean square.

Wide ranging applications of the model (1.1) have been reported particularly in engi-

neering and financial engineering (see e.g. [1],[2], [7],[10]-[13],[16],[25],[27]). Accurate and

efficient solution of these equations are of particular importance in digital simulation, e.g.

for model validation, and in state estimation and control applications. There is an extensive

literature on solutions and discretizations of SDE1 (see e.g. [6],[10j,[18],[22]-[26]). Closed

form solutions were given for a non-linear class of SDEls in [14] by the author. There has

been significantly less work on solutions and discretizations of SDESs. In section 2 the so-

lutions presented in [14] are generalized to jump-diffusions to analytically solve a non-linear

class of SDESs. On methods of numerical solution for jump-diffusion SDEs Maghsoodi

[12] generalized Milshtein's [18] appproach of Taylor expansion of semi-group operators to

jump-diffusions and derived mean square efficient (second order in grid size) discretization

schemes for SDES under weaker conditions (see also Maghsoodi [15]). In many applications

the purpose of the discretization is to approximate E{f(xi)) rather than the process x t

itself. Discretrzations which aim to fulfil this criterion have been referred to as weak ap-

proximations. It is known that these two criteria are far from equivalent (see [12],[19]). This

paper further generalizes the approach of expansions of semi-group operators to the problem

of derivation of new digitally implementable difference schemes for efficient (second-order)

approximation of the solution of equation (1.1) such that :

where ict, denotes the approximator of xtk at the tih point t t of an equidistant partition

with grid size h , and for a > 0, limhlo 9 = 0 for all u E [ O , C Y ) . Furthermore the

algorithms will be constructed in such a way that they are also efficient in the m.s.e. (mean

square error) sense, i.e. :

E[lxt, - ici, 1 2 ] = 0 ( h 2 ) k = 1 ,2 , . . . , M . (1.3)

Hence the resulting algorithms will be doubly efficient. It will be assumed that during [ t o , TI the jump times of the Poisson process and the values of the driving Brownian motion at

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JUMP-DIFFUSION ITO EQUATIONS 105 1

these times as well as at regular sampling points are available. The smallest a-algebra with

respect to which these random variables are measurable will be denoted by P M ( r , T ) . The

use of this information is consistent with on-line filtering applications as well as Mont6Carlo

simulation where sampling of the driving processes a t desired time points is performed by

some hardware or software device.

Consider the class of non-linear jump-diffusion SDEs :

where at , Kt and ct are bounded and continuously differentiable functions of 2. Equation

( 2 . 1 ) is a jump-diffusion generalization of the Extended C I R model ( J D E C I R ) (see Magh-

soodi [14]) . This section presents the exact closed form solution to this class when:

Let the n-dimensional process Yt = (yl(t) , . . . , y, , ( t ) ) ; t 2 0 satisfy:

Theorem 2.1 ( Exact solution of (2 .1 ) ) The patwise unrque solutton of the SDE (2 .1 )

under (2 .2 ) ts gtven b y :

x, = qTy,, ( 2 . 4 )

wzth YoTy0 = 2 0 .

Proof Apply the generalized It8 lemma in [5] to the right hand side of (2 .4 ) while using

(2 .3 ) to obtain:

Given the Brownian motion w, we can choose Wt = ( w l ( t ) , . . . , w , ( t ) ) such that:

is the given Brownian motion driving ( 2 . 1 ) (see [14]) ( I I denotes the Euclidean norm).

Hence p d ~ i = IV,Jdwt. Using this and (2 .2 ) in (2 .5 ) we obtain the dynamics (2 .1 ) as

required.

Corollary 2.1 Under (2 .2 ) the exact closed form solution of ( 2 . 1 ) is given by :

X , = . x P ( - ~ K S d s + 2 l G ~ N . ) I Y O + f /t b 8 ~ ( s , w ) d ~ s / 2 , (2 .7 )

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MAGHSOODI

Proof y,(t), i = 1 , . . . , n form an independent set of processes each the solution of a linear

scalar jump-diffusion SDE (see [5]) and are given by :

~ ( t ) = eXp(- i jll h , d s + ~ t ~ + ~ s ) { Y t ( ~ ) + j l l b s @ ( s . u ) d W ( s ) ) (2.8)

m

Application of the generalized It6 formula shows that the exact first and second mo-

ments, mt and St of the solut~on of (2.1) are given by :

where Et = exp[S,'((2c, + c3) l . - 2KS)ds]. An example of the above non-linear class wdl

also be used to compare the efficiencies of various numerical schemes in sectlon 6.

3. EXPANSIONS IN EXPECTATIONS SEMI-GROUP

To develop the approach for numerical solution let Xt denote the state in the extended

state space El = [O,co) x Rn, resulting from augmenting time to the state x t . Thus (1 1)

can be wrltten as an autonomous SDE3 :

dXt = A(Xt)dt + B(Xt)dWt + C(Xt)dNt, t 15 [to,T], Xt, = Xo w p 1. (3 1)

Let. 81 denote the Banach space of all real valued bounded measurable functions on E l Let

A denote the infinitesimal operator associated with the transition function of the solution of

(3 1 ) wlth domain Z)(A). If f E D ( A m ) 81 then the Taylor Expansion of the Contraction

Senwgroup of Bounded Linear Operators (TECSBLO) can be written as (see [8],[12],[18])

where Ex IS the expectat~on conditional on {w;x, = x) To deal w ~ t h unbounded state

spaces we need to broaden the class of funct~ons and SDE3 coefficients for w h ~ c h f E D(Am)

and (3 2) as well as the formula for the generator are valid. Theorems 3.1 and 3.2 below

provlde sufficient condit~ons

Definzlzon 3 1 (Polynom~al res t r~c t~on) F j w~l l denote the class of funct~ons f [to, TI x

Rn - R, wh~ch are J tlmes contmuously d~fferent~able and for wh~ch there are constants

u, > 0 and Integers p, > 1, r = 0 , 1 , , J such that

l ~ ( ' ) f ( t , x ) l ~ u , ( l + I x l ~ ' ) , r = O , l , , J t E [ t o , T ] , (3 3)

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where D(') denotes any rth order mixed partial differential operator and D ( O ) f f .

Theorem 3.1 ( The weak infinitesimal operator ) Let the coefficients of the SDE3 ( 1 . 1 )

satisfy the untform Lipschztz condition and together with the function f be in class Fg and

let ( 3 . 1 ) have a pathwzse unzque solutzon. Then f E V ( A ) and

where V , f ( s , x ) zs the vector wzth i t h element 3, [xly] denotes the inner product of its a2 vector arguments and V;,f ( s , x ) denotes the matrix with i j 'h element f (s, x).

Proof See Maghsoodi [ 1 2 ] .

Remark 3.1 When the jump coefficient c vanishes identically i.e. S D E 1 , the infinitesimal

operator reduces to the first three terms in ( 3 . 4 ) and is denoted by L . The last term will be

denoted by the operator LC i.e.

(The symbol '2' is used to introduce notations) Hence :

For functins f and g in V ( L ) some algebra leads to the formula

where

and B, denotes the rth row of the matrix B in ( 3 . 1 ) . For the diffusion only case it was

shown in [18] that under boundedness and continuity of a and a and their partial derivatives

of up to the fourth order the TECSBLO formula ( 3 . 2 ) is valid for m = 3 for a class of

functions representing the square of a discretization error. Theorem 3 . 2 below provides

broader sufficient conditions for jump-diffusions:

Theorem 3.2 ( The expansion formula ) If :

1. The function f belongs to the class F7.

2. The coeficients of the SDE3 ( 1 . 1 ) belong to the class F5. 4. The SDE3 ( 3 . 1 ) has a pathwise unique solution (Xi; 31, t E [to, TI)

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5. E[(x~(P'] is un(form1y bounded tn [to,TJ for suficiently large p*'

Then f(.) E V(A3) and the followzng 3Pd order TECSBLO formula 1s valid

Ex,f(Xt,+h) = ( h - r ) 2 E x , A 3 f ( ~ : o + r ) d r . (3.9)

Proof See Appendix B.

s Maghsoodi [12] generalized the semi-group approach to jump-diffusions under polyno-

mial growth. For diffusions Milshtein [20] further characterized the relationship between

the one-step and the k-step weak error under these conditions. In this section these rela-

tions are further generalized in the context of the jump-diffusion semi-group methodology.

Consider an equidistant partition 0 5 to < t l < . . . < tM = T of [to, T] with step length

h = ( T - to)/M and let t E [ to,T - h] and s E ( t ,T] . Let xt,,(s) denote the exact solution

of (1.1) at s given that x ( t ) = x w.p.1.. Suppose at t + h E [to + h , T ] , ~ t , ~ ( t + h) is

approximated by li:,,(t + A ) using x ( t ) = x via the scheme,

where S is an n-vector valued continuous function and is a P M ( ~ , t + h)-measurable

random vect,or independent of x and with bounded moments of sufficiently high orders.

ict,,(t + h) will be referred to as a one-step, approxiinator of the solution of (1.1) (see e.g.

(5.5)). Thus one can construct the (k + 1)-step approximator fth+l from extension and

recursive application of the one-step scheme, i.e.:

where for k = 0, . . . , M - I each Tk+l is PM(T, tl;+l)-meas~rable and independent of

xo, X I , . . . , xk with bounded moments of sufficiently high orders and together they form

an independent set (see e.g. (5.2)). The proof of the following theorem follows very similar

steps to that of Theorem 2 in [20].

Theorem 4.1 Let the sequence {irk) satisfy (4.1) and (4.2) and assume that :

(i) For the jump-diffuston (1.1), E((x(~)(~"') is bounded f o r t E [to,T] and sufi-

ciently large m > 0.

(ti) The function f ( x ) : Rn - R E FZp+2, . p > 0

(izi) For suficiently large m (see (iv)) ~ ( l ? k J ~ " ) etisls and is uniformly bounded

for all k = 0 , 1 , . . . , M and M E N.

a 1. p* 2 I l p + 6pZ + 2p3 + 2p4, where p = maxoskss ph (see [12]).

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JUMP-DIFFUSION ITO EQUATIONS

( iv) 3 K ( x ) : Rn -r R in Fo with po < 2m such that Vt E [to> T - h] :

then for all k = 0 , 1 , . . . , M and M E N there ezisls a constant L1 > 0 such that:

i.e. scheme (4.2) is of O(hp) in the weak sense.

Thus to derive a second order weakly efficient scheme it may be sufficient to construct

a weakly third order one-step approximator. The validity of condition (iii) of Theorem (4.1)

however depends on the structure of the scheme (4.2). Sufficient conditions on the structure

of (4.2) were given in [20] for a class of SDEl schemes which do not make explicit use of

the increments of the Brownian motion. Lemma 4.1 below provides sufficient conditions so

that the jump-diffusion schemes of this paper can make explicit use of these increments.

Definition 4.1: ( Regular schemes) If a jump-diffusion approximation scheme satisfies

(4.1) and (4.2) and there exist constants L3 > 0 and L4 > 0 and a real valued function

L5(h . Tk+]) such that for h < 1 :

then the scheme will be said to be a Regular scheme

Lemma 4.1 If the sequence { j i b ) is generated b y a regular approzinlalion scheme then thr

result of Theorem 4.1 will still hold without condition (iii).

Proof From (4.2) we have :

Taking expectations and using (4.5) for the second term in (4.8) we have:

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Similarly for j = 2, . , . , 2 m using (4.6) and (4.7) we have:

Thus it follows from (4.8)-(4.10) that for positive L6 depending on m only:

Thus iterating the R.H.S. of (4.12) through k = M - 1 , . . . , 0 and noting that h = ( T - t o ) / M

we find that in any regular scheme E ( ( f k lZm) exists and is uniformly bounded for all m and

k 5 M in N which is condition (iii) of Theorem 4.1. as required.

5. DERIVATION OF THE ALGORITHMS

Consider the general scalar SDE3 :

The basic stochastic Cauchy-Euler (SCE) discretization scheme for this equation is:

A where the hatted functions denote their evaluations at the point (tk, XE) , Awk+1 = wt,+, - A

wt, and ANk+, = Ntk+, - Ntk. The best m.s.e. achievable by PM(T)-measurable diffusion

schemes is of O(h2) [3]. This result carrles over to jump-diffusion PM(r,T)-measurable

schemes. Maghsoodi [12] showed that the scheme (5.2) is only of O(h) in m.s.e. and

developed jump-diffusion semigroup methods to derive algorithm Z ((5.3) below) and prove

that it is of O(hZ) in m.s.e. :

Algorilhm Z:

1 * * 1 * 1 " >

ik+l = x k + (2 - ;aa,)h 2 + bAwk+l + -ba,Aw;+, + -(3c - C , ) A N ~ + ~ 2 2 1 . + (b, - ~ ) A w ~ + ~ A N ~ + ~ + $cc - E)AN:+~ + (b& - be + B ) A z ~ + ~ ,

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JUMP-DIFFUSION ITO EQUATIONS 1057

A where the notation is as in (5.2) in addition 6, i %(tk,ik), Be = b( tk rzk + &) and

AZk+l fi ~ ' ~ " ( w , - ws) dN,. Theorem 5.2 below shows that both of the schemes SCE and

Z are in fact only of O(h) in the weak sense i.e, if the sequence {xk} is generated by (5.2)

or (5.3) then :

IE[f(zk) - f ( i t ) ] l = O(h) k = 1,2, . . . , M. (5.4)

In this section alternative jump-diffusion algorithms will be derived and proved to be efficient

in the weak sense (1.2) as well as in m.s.e. In the light of Theorem 4.1 the first step of the

iterations will be focused on. The idea is to extend the structure of (5.3) by including

additional PM(r , to + h)-measurable random variables i.e. :

For this one-step scheme to be of 3rd order in the weak sense it may be sufficient that :

Consider the augmented SDE3 systems :

(i) dXl(t) = s , z,)ds + a(s , z,)dw, + c(s, x , ) d N , dt = dt,

dwt = dwi dNt = dN* dZi = (wt - wi,)dNt dt = dt:

where Xl(to) = ( x ~ , t o ) ~ and X2(t0) = (wtor NtorZtorlO)T. Let A1 and A1 denote the gen-

erators associated with systems (5.7)(i) and (5.7)(ii) respectively. We can write f ( i ta+h) = f (9(Xz( to + h)) where the function O(.) is given by the right hand side of (5 5). Thus ~f

the conditions of Theorem 3.2 are satisfied we can write the TECSBLO formulae associated

with the above augmentea systems as :

1 1 Ef( i1) = g(X2(fo)) + ~ ~ A ~ s ( X Z ( ~ O ) ) + j h2Aig (~z ( t a ) ) + j J ~:E-&(xdto + r))dr.

f 0

A A (5.9)

where g = f o O and A, = h - T . Furthermore under the conditions of Theorem 3.2 the

expectation integrands are uniformly bounded hence both remainders are of O(h3). Thus

for the scheme (5.5) to satisfy (5.6) it is sufficient that :

( i ) Alf(zo) = Azf(;i.o), (ii) A:f(zo) = Aif( io) . (5.10)

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1058 MAGHSOODI

These equations can be evaluated by applying formulae (3.6)-(3.10) to the augmented state

dynamics (5.7). We have :

where the subscripts @, t and w denote the corresponding partial derivatives. The right

hand side of (5.12) can be further expanded in terms of @(Xz(to)), @c(X2(to)) and their

partial derivatives by using formulae (3.6)-(3.10). For instance :

where s@ = @,. Similar calculations lead to the following representations of equations

(5.10) in terms of the coefficients cl -cia :

where f ( P ) denotes the pih derivative o f f with respect to x and unless otherwise specified

all the functions are evaluated at the points (xo) and (to, xo). For (5.14) and (5.15) to hold

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JUMP-DIFFUSION ITO EQUATIONS 1059

independently of the choice of f and At, it is sufficient that the coefficients of f ( ' ) - f ( 4 )

and At,, X I : ) , A!:) on both sides are equivalent. This leads to the solution set :

1 1 c1 = a - -oa,

2 C2 = u

C3 = 2uux 1 1 1

cq = -(3c - c,) cg = - ( c , - c) cg = - 2 2 2 [ L ~ c + a , - a1 1 1

cs = - ( f l u + ua,) C I O = -Lla (5.16) 2 2

The coefficients cs and c7 can now be chosen such that the resulting scheme has exactly

the same coefficients for each of the seven terms of the scheme Z which it contains. This

choice will yield an algorithm which is doubly efficient. Substituting the solution back into

the scheme structure (5.3) and extending the recursion to the (k + I)" step we obtain:

Theorem 5.1 ( Algorithm OP is efficient in the weak sense) If the sequence {it) satisfies

(5.17) and the conditions of theorems 3.1 and 3.2 together with condition (iii) of Theorem

4.1 hold, then :

IE[f(zt) - f ( ~ ~ ) ] l = O(h2) k = 1 , 2 , . . . , M . (5.18)

Proof By hypotheses conditions (i)-(iii) of Theorem (4.1) are satisfied. Condition (iv)

follows from the construction of the scheme since the conditional versions of formulae (5.8)

and (5.9) equally hold for to replaced by t E [ to, T - h] and zf, and i t , replaced by x Equations (5.14) and (5.15) also hold for these values. Thus the left hand side of inequality

(4.3) is the difference of the integral error terms in the expansions. The integrand in (5.8)

would now contain :

for sufficiently large rn and where the first inequality follows from the conditions of theorems

3.1 and 3.2 and the second inequality follows from conditional version of (A.6) (see Appendix

A ) . Similarly the corresponding integrand in (5.9) involves : Dow

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1060 MAGHSOODI

where the the functions cF CC,(z) denotes an upper bound of the sum of the Ft-measurable

coefficients in the scheme and G,(@) denotes that of the mixed partial derivatives and shifts

present in dig (see e.g. (5.15)) and these functions are in Fo. Hence :

where L15(x) E FO with p0 = 2m The proof of the theorem now follows since condition

(iv) of Theorem (4.1) holds. 1

Corollary 5.1 If the condztzons of theorems 3.1 and 3.2 hold and the functzons appeartng

zn the srheme OP are unzfrom Lzpschztz conlanuous then OP zs a regular scheme and I S stdl

of O(h2) In the weak sense

Proof The scheme (5 17) satisfies conditions (4 1) and (4 2) Cond~tlon (4 5) follows

from taklng expectation from the R H S of (5 17) and the unlform Llpschltz continuity

Cond~tions ( 4 6) and (4 7) follow from uniform Lipschitz continuity and that the function

Ls(h, Tk+l) can be formed by summmg all the random variables and the h terms on the

R H S of (5 17) all the moments of orders j = 2, , 2 m of whlch are of O(h) for h < 1

Hence (4 20) is a regular scheme and by Lemma (4 1) E(2zm) 1s uniformly bounded Thus

by Theorem 4 1 the scheme OP is of O(h2) in the weak sense 1

Theorem 5.2 (SCE and Z are inefficient in the weak sense) If the sequence {xk) rs grven L

b y ezther of the schemes SCE (5 2) or Z (5 3) , then under the condtttons of Theorem 5 1

IE[f(xk) - f ( lk)] l = O(h) k = 1,2 , , M Furihrrrnore, these schemes cannot achteve

any hzgher order zn the weak sense.

A Proof For 0 5 s < 1 5 T let u (s ,x ) = Ef(xr /x , = z) . Under the conditions of Theorem i

5.1 we can write :

t - *

~ ( s . Z) = f ( ~ ) + 11 E[AJ(x9 + T ) / x ~ = z ] d ~ . (5.19)

Smce dl f E F5, f E F7 and the jump-diffusion (5.7)(i) has uniformly bounded moments if

x does (see A.6, Appendix A) therefore u E F7 . Thus we can write (see also [20]) :

The functions f and u satisfy the conditions of theorems (3.1) and (3.2) and 2, has bounded

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JUMP-DIFFUSION ITO EQUATIONS

moments o f 2mfh order. Hence we can apply the expansions (

1061

5.8) and (5.9) t o obtain :

(5.21)

where for the scheme 2, dz is the generator o f the process X 2 ( t ) (5.7)(ii) with O given

by the R.H.S. o f (5.3) and for the scheme SCE, X z ( t ) lacks the third component and O is

given by the R.H.S. o f (5.2). By evaluating equation (5.14) for the function u at the points

( t , , xi.,) for i = 1,. . . , k - 1 and for the function f at the point ( t k , ik) we find that it holds

independently o f the choice o f u and f for both schemes SCE and Z (see also [12]). However

the second order equations (5.15) cannot hold for these schemes since for instance unlike

the scheme OP they do not contain the term Lla appearing in the R.H.S o f (5.15) in their

structure Hence:

k - 1 1 E [ f ( x k + ~ ) - f ( i k+ l ) ] = E ~ { ~ h ~ [ d : u ( t ~ , i r ) - d i u ( t i , @ ( ~ , ( t , ) ) ) ] + o ( h 3 ) )

i = l

The square bracket terms on the R.H.S. o f (5.22) are non-zero hence for arbitrary k < M E N

the R.H.S. o f (5.22) is o f O ( h ) and this order is the best that can be achieved by these schemes

in the weak sense.

Remark 5.1 I f in addition to the conditions of theorems (3.1) and (3.2) the coefficients

appearing in the SCE or Z algorithms are uniform Lipschitz continuous then these schemes

become regular and Theorem 5.2 would be valid without any additional conditions.

Theorem 5.3 (OP is efficient in the m.s.e. sense) If the sequence ( 2 k ) is given by (5.17)

then under the conditions of Theorem 5.1 we have E [ ( z k - ~ k ) ~ ] = O ( h 2 ) , k = 1,2, . . , , M .

The following lemma is the extension o f the diffusion result, in [21] to jump-diffusion semi-

group methods o f this paper and will be used to prove Theorem 5.3. The proof o f Lemma

5.2 can be found in Maghsoodi [15].

Lemma 5.2 Let the sequence { i r k } of the approzimalors of the jump-diffusion (1.1) satisfy

(4.1), (4.2) and condition (iii) of Theorem 4.1 and be of O(hP1) in absolute mean error and

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1062 MAGHSOODI

of O(hP2) tn mot mean square error tn one step. More preczsely suppose Vt E [ t o , T - h] and

x E Rn, 3 functtons L s ( x ) > 0 and L ~ ( x ) > 0 zn F o such that:

1 1 [ E / X ~ , . ( ~ + h ) - iicqX(t + h)12] ' 5 L,(x)hP2 p2 2 j, P I > p~ + j, (5.24)

then 3La > 0 such that for all M E N and k = 0, . . . , M - 1

i.e. the scheme generating the sequence {fir;} zs of O(h2P) in m.s.e. with p = p2 - ' 2 '

Proof of Theorem 5.3 By Lemma (5.2) it is sufficient to prove that the O P algorithm A

satisfies (5.23) and (5.24) with pl = 2 and p2 = $. Let @ ( X ( t + h ) ) = z ( t + h ) - i ( t + h)

where x(t+h) is generated via scheme O P and X denotes the process satisfying the dynamics

resulting from augmentation of (5.7)( i) to (5.7)( i i ) with generator A. Thus by the arguments

used in the proof of Theorem 5.1, for r = 2 , 3 , E [ A ' @ ( X ( 1 + r ) ) / X ( t ) = X ] appearing in the

integrand error terms of formula (3.9) are bounded by functionals of the scheme coefficients

evaluated at z (see proof of Theorem 5.1). Hence we can write:

I E , 0 2 ( x ( t + h ) ) = @'(x) + h A a 2 ( X ) + 2 h 2 A 2 @ 2 ( X ) + L 1 0 ( ~ ) 0 ( h ~ ) , (5.27)

where L 9 ( x ) and L l o ( z ) are in F o with p, = 2m for some finite m. We have @ ( X ) G 0 and

application of formulae (3.4) and (3.8) to the new augmented jump-diffusion process gives:

Now by (3.5) and (3.6) :

Using (5.28), (5.29) and (3.5) we obtain:

Thus using (5.30) in (5.26) and (5.27) we find that the scheme O P satisfies the conditions

of Lemma 5.2 with pl = 2 and p2 = $. Hence this algorithm is of second order in the m.s.e.

sense i.e. algorithm O P (5.17) is doubly efficient. rn

Remark 5.2 If the coefficients in scheme O P are uniform Lipschitz continuous the scheme

O P becomes regular and Theorem 5.3 would only require the conditions of theorems 3.1

and 3.2.

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JUMP-DIFFUSION ITO EQUATIONS 1063

Remark 5.3 ( Second order Jump-adapted schemes) The method of jump-adaption which

involves the incorporation of the jump times into the regular partition (see also [17]) can

lead to simpler schemes. Below the doubly efficient jump-adapted algorithm (OPJA) and

an alternative class of doubly efficient direct jump-adapted schemes (OPDJA) are derived.

Let 0 5 to = TO < TI < . . . < TN = T denote the new augmented partition of [to, TI. Thus - T ~ ) 5 h w.p.1. Taking left continuous sample paths the interval

[ rk , T ~ + ~ ) , k = 1 . 2 , . . . , N can only contain AN,,,, = 0 or 1 jump. Using this property in

the scheme OP and simplifying we obtain :

Algorithm OPJA:

A . A where now i k = z r k , the hatted functions are evaluated at (rk, i k ) , = r h + l - rk, and

A Aw,,,, = w,,,, - w,, . The results obtained for the scheme OP also hold for the scheme

OPJA.

A class of alternative doubly efficient jump adapted schemes can also be derived by

noting that the path of the solution between jump times is that of the corresponding diffusion

restarting from the position immediately after the last jump. Suppose Slp is an SDEl

scheme of order @with the general structure :

then the corresponding direct jump-adapted SDE3 scheme S3@ can be written as:

Thus the doubly effieient direcf jump-adoptedSDE3 scheme which also generalizes the weakly

second order SDEl scheme in [19] to SDE3s is :

Algorilhm OPDJA:

A where the notation is as in (5.31) in addition Si = a(rk,zk + ? A N , , + , ) . In practice and

partriculary for nonlinear equations with high jump intensities jump-adapted schemes may

suffer from computational inefficiencies due to excessive function evaluations a t the points

(rk,ik + EANrktl).

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1064 MAGHSOODI

Remark 5.4 ( The n-dimensional doubly efficient scheme) The methods employed in the

scalar case can be analogously applied to derive a discretization scheme for the system of

SDE3s (1.1) :

Algorzthm OP2:

where ai, denotes the i th column vector of a and V,ur. is the matrix with the i j t h element

8u,,/8xj. All the above results for the scalar scheme OP carry over to the n-dimensional

scheme OP2 subject to the additional commutativity condition V,uj. u,. = Vxu,. u j . for

j , r = 1 , . . . , q . (see also [12] and [24] ). (5.35) is doubly efficient i.e. it satisfies (1.2) and

(1.3) simultaneously. The jump-adapted OP2JA and the direct jump-adapted OP2DJA

versions of (5.35) can be written in analogy to (5.31) and (5.34) respectively.

Remark 5.5 (A doubly efficient scheme for SDE1) If the jump component in algorithm OP2

(5.41) is set to zero we obtain a new scheme (see also Maghsoodi [12]) for the multivariable

SDEl which is doubly efficient under the conditions of Remark 5.4 :

Algorithm OP12:

Remark 5.6 ( Random jump sizes ) The extension of the above methods and schemes to

equations with random jump sizes is also quite analogous. The third integral in (1.1) takes

the form J, c(s, x,, u)N(ds, du) and it is now with respect to the inhomogeneous integer

valued Poisson random measure {N([t,, t ) ,T); F t , t E [t , ,T])} , representing the number of

jumps occurring during [to, t) with random jump sizes which have possible values in the A

set T in the u-algebra of the Bore1 sets B(U) of U = Rn\{O). For each r, N([to,t) ,T) has the Poisson distribution with parameter E{N([to,t), r)) = ?r(s,T)ds < co where

a(s , T) is the intensity rate at time s E [to, T] of the jumps whose size is in r and for each

s E [to,T], ~ ( s , .) is also a probability measure on 2.4 representing the jump size density.

The operator C, introduced In (3.5) now takes the form :

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The calculations and the results carry over conveniently to yield the random jump analogue

of the scheme OP2:

Algorithm OP2R:

A where the notation is as in (5.35) and = [tk,tk+l). Likewise the solutions presented

in section 2 for the non-linear class (2.1) are easily modified for the random jump case

by replacing ct with c(t ,u) and appending the integrals w.r.t. u over U in all the terms

containing it.

6-

The computational performance of the proposed schemes can be compared with those

of the schemes Z and SCE by comparing their efficiencies in simulating the moments of

the solutions of numerical examples. The mean and variance of the solution process can

be estimated from a number of sample paths simulated via each of the schemes and the

estimation errors can be computed using the known exact moments and compared. In each

simulation trial the same sequence of pseudo-random samples from the driving processes

were used for all the schemes. Consider the example :

with K = KO = 0.4, a = 0.5, c = 0.2, zo = 2.5 and Poisson intensity rate At = 3t + 3

Equation (6.1) is an example of the JDECIR class (2.1) which were solved exactly in section

2. The choice of this example is governed by the requirements of non-linearity, availability

of the exact solution as well as the exact mean and variance and the dependence of the

jump term on the state. The mean number of jumps during [0,2] is 12. As observed in

the sample paths (FIG. 1 ) the jump intensity and the magnitude of the jumps affecting

the paths increase over time. The solution is second moment asymhotically unstable and

its rapid growth is mainly due to large positive jumps. These features test also the ability

of the scheme OP (5.17) in capturing large jumps robustly and efficiently. The formula :

Eficrency = (Constant) x (Average error x Computrng itme)-' is used. Figure 1 displays

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1066 MAGHSOODI

" 0 0.2 0 4 0.6 0.8 1 1.2 1.4 1.6 t 1 . 8 2

FIG. 1

Exact mean & variance and two simulated paths

FIG. 2 Exact mean & variance and their estimates via scheme SCE

the exact mean and variance together with two simulated sample paths of the solution of

(6.1). Figures 2-4 display the exact mean and variance together with their estimates based

on 16 grids and a sample of 10000 simulated paths generated by the schemes SCE. Z and

OP respectively. CPU times ranged between 10.27 and 15.93 seconds on a Pentium 90.

Average absolute estimation errors over the interval [O, 21 were used to compute efficiencies

for each of the three schemes. The simulation results are consistent with the theoretical

findings. For instance trials with 10 and 16 grids showed that in estimation of the mean the

OP scheme was more efficient than SCE by 511% and 1070% respectively and in estimation

of variance by 669% and 1100%.

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FIG. 3 Exact mean & variance and their estimates via scheme Z

FIG. 4

Exact mean & variance and their estimates via scheme OP

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REFERENCES

MAGHSOODI

[l] Au, S. P., A . H. Haddad and H. V. Poor: A state estimation algorithm for linear systems driven simultaneously by Wiener and Poisson processes. IEEE Trans. Automat. Contr., Ac-2J, Nb. 3, 1982, 617-626.

(21 Bodo, B. A, , M. E. Thomson and T . E. Unny: A review of stochastic differential equations for applications in hydrology. J. Stoch. Hydro]. Hydraulics, 1, 1987, 81-100.

[3] Clark. J . M. C. and R. J . Cameron: The maximum rate of convergence of discrete approximations for stochastic differential equations. In: B. Grigelionis, Ed., Lecture Notes in Control and Informatron Sciences, z, Springer-Verlag, Berlin, 1980.

[4] Dynkin, E. B.: Markov Processes. I , Springer-Verlag, Berlin, 1965

[5] Gikhman. I. I. and A. V. Skorokhod: Stochastic Differentral Equations. Springer-Verlag, Berlin, 1972.

[6] Golec, J . and G. Ladde: Euler-type approximation for systems of stochastic differential equations. Journal of Applied Mathematics and Stmulation, 2, Nb. 4 , 1989, 239-249.

[7] Harris, C. J.: Stability and control of flexible spacecraft with parametric excitation. Oxford IJnivcrsity Eng. Lab. Research Report Nb. 1193/77, 1977.

[8] Hille, E. and R. S. Phillips: Functional Analyszs and Semi-groups. Providence, R.I. Amer. Math. Soc., 1957.

[9] It6, K.: On stochastic differential equations. Mem. Amer. Math. Soc., 4, 1951, 1-51

[lo] Kloeden, P. E. and E. Platen: Numerical Solution of Stochastic Differential Equations. Springer-Verlag, Berlin, 1992.

[I l l Kushner, H. J . and G . DiMasi: Approximations for functionals and optimal control problems on jump-diffusion processes. J. Math. Anal. Appl., a, 1978, 772-800.

[12] Maghsoodi, Y. : On approximate integration of a class of stochastic differentla1 equa- tions. D. Phil. Thesis, Eng. Science Dep. Oxford University, 1983.

[13] - : Time varying jumps in stock prices and a new option pricing formula. Working Paper Nb. O.R. 39, Dep. Math., Southampton University, 1992.

[14] - : Solution of the extended CIR term structure and bond option valuation. Math- ematiral Finance, 6, 1996, 89-109.

[15] - : Mean square efficient numerical solution of jump-diffusion stochastic differential equations. Sankhya: The Indian Journal of Slalistics, 58, Series A, Pt . 1 , 1996, 25-47.

[16] Maghsoodi, Y. and C. J . Harris: In-probability approximation and simulation of non- linear jump-diffusion stochastic differential equations. IMA J. Math. C0ntr .b Inf., 4, 1987, 65-92.

[17] Mikulevicius, R. and E. Platen: Time discrete Taylor approximations for It6 processes with jump component. Math. Naehr., a, 1988, 93-104.

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JUMP-DIFFUSION ITO EQUATIONS 1069

[18] Milshtein, G . N.: Approximate integration of stochastic differential equations. Th. Prob. Appl.. u, 1974, 557-563.

[19] - : A method of second-order accuracy integration of stochastic differential equa- tions. Th. Prob. Appl., 2, 1978, 396-401.

[20] - : Weak approximations of solutions of systems of stochastic differential equations. Th. Prob. Appl., a, 1985, 750-766.

[211 - : A Theorem on the order of convergence of mean square approximations of solutions of systems of stochastic differential equations. Th. Prob. Appl., 2, 1988, 738-741.

[22] Newton, N. J . : An asymptotically efficient difference formula for solving stochastic differential equations. Stochastics, @, 1986, 175-206.

[23] Rao, N. J . and J. D. Borwankar and D. Ramkrishna: Numerical solutions of Ito integral equations. SIAM J. Control, Q, Nb. 1, 1974, 124-139.

1241 Riimelin, W.: Numerical treatment of stochastic differential equations. SIAM J. Numer. Anal., 19, 1982, 604-613.

[25] Snyder, D. L.: Random Poznt Processes. Wiley, New York, 1975.

[26] Talay, D.: Simulation and numerical analysis of stochastic differential systems. In: KrBe, P. and W. Wedig, Eds., Effective Stochastic Analysis., Springer-Verlag, Berlin, 1988.

1271 Zhang, X . L.: American options and jump-diffusion models. Comptes Rendus De LJAcademie Des Sciences, Serie I-Mathematique, 9, 1993, 857-862.

If u , is an adapted measurable process such that:

then :

E [i''h u , ~ z u , ] ~ ~ 5 [ m ( 2 m - l)lmhrn-' l:oth E ( U : ~ ) ds h 5 T - t o . ( A . 2 )

(see [5]) . This result was generalized to It6 integrals with respect to inhomogeneous Poisson

counting process by Maghsoodi [12] :

Lemma A . l If u , is an adapted measurable process such that:

r T

then 3 a constant cp > 0 such that:

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1070 MAGHSOODI

For SDES (1.1) if there exists a constant L > 0 such that for all t E [to. T ] and x E Rn

and (1.1) has a pathwise unique solution in [to,T] with E(lxt12'p) < OC, for integers r 2 1

and p 2 1, then there exist constants L l , p > 0 and Lz, , > 0 such that:

(see [12] ). Application of Holder's inequality in SDE3 yields

APPENDIX B. Proof of Theorem 3.2

An idea suggested for diffusions in [18] is generalized to jump-diffusions and imple-

mented. Assume without loss of generality that @ 2 0 (see [12]) and define the sequence

{aN; N E N ) as follows

(B.1) where L N ( @ ) is the Lagrange interpolation polynomial such that

where LC)( . ) denotes the i th order derivative of L N ( . ) . For all N E N the functions @ N ( X ) :

El C R+xRn - R a r e bounded and 7 times continuously differentiable with bounded mixed

partial derivatives. Hence by Theorem 3.1 c P N E D ( d 3 ) . Therefore the following third order

TECSBLO formula is valid:

It is shown that when N tends to infinity (B.3) tends to the corresponding TECSBLO

formula for a. Starting from the left hand side of (B.3), since limN,, @ N ( X ~ , + ~ ) = @(Xto+h) w.p.l . , by dominated convergence theorem

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JUMP-DIFFUSION ITO EQUATIONS 1071

Clearly Q N ( X o ) - @ ( X o ) as N m. To show that limN,, A @ N ( X O ) = d @ ( X n ) , by

(3.4) we can write

1 A @ N ( X O ) = Z t ~ { [ ( @ ~ ) ( 2 ) ( ~ ~ @ ) T ( V ~ @ ) + ( @ N ) ( ' ) V X X @ ] B B ~ )

+ [ (@N)(~)vx@(xo)JA(xo)] + & [ @ N ( X O + C O ) - @N(XO)lr (8.4)

where ( @ N ) ( ~ ) denotes the i t h order derivative of aN with respect to @. We have

lim @N ( X O + C O ) = @ ( X O + GO), N]JII(@N)(') = 1, N - m

and lim ( @ N ) ( ' ) = 0 s > 1. N - m

Using (B.5) and (B.6) in the R.H.S. of (B.4) and comparing with (3.4) we obtain

lim d G N ( X o ) = A @ ( X o ) . N - m

Now to prove convergence of A 2 a N ( X o ) , by (3.6) we have:

d 2 @ ~ = L'@N + CLC@N + L c L @ ~ + c;@N.

From (0.5) and (B.6) it follows that:

L in (B.8) is a differential operator thus by (B .9) the diffusion-only terms in (B.8) converge.

Also by (B.9):

lim @:)(xo + Ca) = 0 1 < s < 7 , (B .11) N- m

and this limit is 1 for s = 1. The limits of the jump-diffusion terms in (B.8) involve combined

expressions such as limN,, LION(Xo + C o ) - @ J ~ ( X ~ ) ] . Thus by (B.9)-(B.11) these terms

converge. Hence :

lim d2@~(x0) = d 2 @ ( X o ) . (B .12) N-m

d3@N(Xt ,+r) involves polynomiah in terms such as ( @ N ) ( ' ) , @$I, s = 0 , . . . , 6 ; ( @ N ) c ,

( @ N ) ~ ~ , their partial derivatives and similar terms in the coefficients A , B, and

C of the SDE3 (3.1). Applieation of (B.7)-(B.lO) and the arguments used for (B.lO) show

that:

lim d 3 @ ~ ( X t , + r ) = A3@(Xto+,) w.p.1. ( 8 . 1 3 ) N-m

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1072 MAGHSOODI

By polynomial growth restrictions there exists a polynomial in components of Xto+r and

with finite expectation which dominates d 3 0 N ( X t , + r ) for every N E N. Hence by domi-

nated convergence theorem

lim E [ d 3 @ N ( ~ t , + r ) / ~ o ] = ~ [ d ~ @ ( x t ~ + T ) / ~ o ] N-m (8.14)

Since by growth restrictions and (A.6) both sides of (B.14) are bounded, by bounded con-

vergence theorem

J$w l h ( h - T ) ~ E [ ~ ~ @ N ( X ~ . + ~ / O I = ( h - T ) ~ E [ A ~ @ ( X + ) / X ~ ] ~ T . (B.LS)

m

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