Electronic copy available at: http://ssrn.com/abstract=2523110

Long Term Risk: A MartingaleApproach

Likuan Qin∗ and Vadim Linetsky†

Department of Industrial Engineering and Management SciencesMcCormick School of Engineering and Applied Sciences

Northwestern University

Abstract

This paper extends the long-term factorization of the pricing kernel due to Alvarez andJermann (2005) in discrete time ergodic environments and Hansen and Scheinkman (2009) incontinuous ergodic Markovian environments to general semimartingale environments, with-out assuming the Markov property. An explicit and easy to verify sufficient condition isgiven that guarantees convergence in Emery’s semimartingale topology of the trading strate-gies that invest in T -maturity zero-coupon bonds to the long bond and convergence in totalvariation of T -maturity forward measures to the long forward measure. As applications,we explicitly construct long-term factorizations in generally non-Markovian Heath-Jarrow-Morton (1992) models evolving the forward curve in a suitable Hilbert space and in thenon-Markovian model of social discount of rates of Brody and Hughston (2013). As a fur-ther application, we extend Hansen and Jagannathan (1991), Alvarez and Jermann (2005)and Bakshi and Chabi-Yo (2012) bounds to general semimartingale environments. WhenMarkovian and ergodicity assumptions are added to our framework, we recover the long-term factorization of Hansen and Scheinkman (2009) and explicitly identify their distortedprobability measure with the long forward measure. Finally, we give an economic interpre-tation of the recovery theorem of Ross (2013) in non-Markovian economies as a structuralrestriction on the pricing kernel leading to the growth optimality of the long bond and iden-tification of the physical measure with the long forward measure. This latter result extendsthe interpretation of Ross’ recovery by Borovicka et al. (2014) from Markovian to generalsemimartingale environments.

1 Introduction

The stochastic discount factor (SDF) assigns today’s prices to risky future payoffs at alternativeinvestment horizons. It accomplishes this by simultaneously discounting the future and adjustingfor risk (the reader is referred to Hansen (2013) and Hansen and Renault (2009) for surveys ofSDFs). One useful decomposition of the SDF that explicitly features its discounting and riskadjustment functions is the factorization of the SDF into the risk-free discount factor discounting

∗likuanqin2012@u.northwestern.edu†linetsky@iems.northwestern.edu

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Electronic copy available at: http://ssrn.com/abstract=2523110

at the (stochastic) riskless rate and a positive unit-mean martingale encoding the risk premium.The martingale can be conveniently used to change over to the risk-neutral probabilities (Coxand Ross (1976a), Cox and Ross (1976b), Harrison and Kreps (1979), Harrison and Pliska (1981),Delbaen and Schachermayer (2006)). Under the risk-neutral probability measure risky futurepayoffs are discounted at the riskless rate, and no explicit risk adjustment is required, as it isalready encoded in the risk-neutralized dynamics. In other words, the riskless asset (often calledriskless savings account) serves as the numeraire asset under the risk-neutral measure, makingprices of other assets into martingales when re-denominated in units of the savings account.

An alternative decomposition of the SDF arises if one discounts at the rate of return on theT -maturity zero-coupon bond. The corresponding factorization of the SDF features a martingalethat accomplishes a change to the so-called T -forward measure (Jarrow (1987), Geman (1989),Jamshidian (1989), Geman et al. (1995)). The T -maturity zero-coupon bond serves as thenumeraire asset under the T -forward measure.

More recently Alvarez and Jermann (2005), Hansen et al. (2008), and Hansen and Scheinkman(2009) introduce and study an alternative factorization of the SDF, where one discounts at therate of return on the zero-coupon bond of asymptotically long maturity (often called the longbond). The resulting long-term factorization of the SDF features a martingale that accomplishesa change of probabilities to a new probability measure under which the long bond serves as thenumeraire asset. Combining the risk-neutral and the long-term decomposition reveals how theSDF discounts future payoffs at the riskless rate of return, a spread that features the premiumfor holding the long bond, and a martingale component encoding an additional time-varyingrisk premium. Alvarez and Jermann (2005) introduce the long-term factorization in discrete-time, ergodic environments. Hansen and Scheinkman (2009) provide far reaching extensions tocontinuous ergodic Markovian environments and connect the long-term factorization with pos-itive eigenfunctions of the pricing operator, among other results. Hansen (2012), Hansen andScheinkman (2012a), Hansen and Scheinkman (2012b), Hansen and Scheinkman (2014), andBorovicka et al. (2014) develop a wide range of economic applications of the long-term factoriza-tion. Bakshi and Chabi-Yo (2012) empirically estimate bounds on the factors in the long-termfactorization using US data, complementing original empirical results of Alvarez and Jermann(2005).

Contributions of the present paper are as follows.

• We give a formulation of the long-term factorization in general semimartingale environ-ments, dispensing with the Markovian assumption underpinning the work of Hansen andScheinkman (2009). We start by assuming that there is a strictly positive, integrablesemimartingale pricing kernel S with the process S− of its left limits also strictly positive.The key result of this paper is Theorem 4.1 that specifies an explicit and easy to verifysufficient condition that ensures that the wealth processes of trading strategies investingin T -maturity zero-coupon bonds and rolling over as bonds mature at time intervals ofduration T converge to the long bond in Emery’s semimartingale topology. Furthermore,we prove that, under our condition, the resulting probability measure is the limit in to-tal variation of the T -maturity forward measures. We thus call it the the long forwardmeasure.

• As an application, we verify our sufficient condition in a class of Heath, Jarrow and Morton(1992) models with forward rate curves taking values in an appropriate Hilbert space anddriven by a (generally infinite-dimensional) Wiener process and obtain explicitly the long-

2

term factorization of the pricing kernel in non-Markovian HJM models. This applicationillustrates the scope and generality of our results relative to the original results of Hansenand Scheinkman (2009) focused on ergodic Markovian environments.

• As another example, we verify our sufficient condition and obtain explicitly the long-termfactorization in the non-Markovian model of Brody and Hughston (2013) of social discountrates.

• As a further application, we extend Hansen and Jagannathan (1991), Alvarez and Jermann(2005) and Bakshi and Chabi-Yo (2012) bounds to general semimartingale environments.

• Furthermore, when we do impose Markovian and ergodicity assumptions in our framework,we recover the long-term factorization of Hansen and Scheinkman (2009) and explicitlyidentify their distorted probability measure associated with the principal eigenfunction ofthe pricing operator with the long forward measure. Our sufficient condition for existenceof the long bond and, thus, for existence of the long-term factorization, when restrict-ed to Markovian environments, turns out to be distinct from the conditions in Hansenand Scheinkman (2009). Interestingly, it does not require ergodicity, as in Hansen andScheinkman (2009). Indeed, we provide an explicit example of an affine model with ab-sorption at zero, where the underlying Markovian driver is transient, but the long-termfactorization nevertheless exists.

• Viewed through the prism of our results in semimartingale environments, the issue ofuniqueness studied in Hansen and Scheinkman (2009) does not arise, as the long-term fac-torization is unique by our definition as the long term limit of the T -forward factorizations.

• We give some further explicit sufficient conditions for existence of the long-term factor-ization in specific Markovian environments that are easier to verify in concrete modelspecifications than either our general sufficient condition in semimartingale environmentsor sufficient conditions in Markovian environments due to Hansen and Scheinkman (2009).

• As an immediate application, we give an economic interpretation and an extension of theRecovery Theorem of Ross (2013) to general semimartingale environments (from Ross’ orig-inal formulation for discrete-time, finite-state irreducible Markov chains and Qin and Linet-sky (2014) formulation for continuous-time Markov processes). Namely, we re-interpretRoss’ main assumption as the assumption that the long bond is growth optimal. This thenimmediately results in the identification of the physical measure with the long forward mea-sure. This extends the economic interpretations of Ross’ recovery offered by Martin andRoss (2013) in discrete-time Markov chain environments and by Borovicka et al. (2014) incontinuous Markovian environments to general semimartingale environments.

The main message of this paper is that the long-term factorization is a general phenomenonfeatured in any arbitrage-free, frictionless economy with a positive semimartingale pricing kernelthat satisfies the sufficient condition of Theorem 4.1, and is not limited to Markovian environ-ments.

The rest of this paper is organized as follows. In Section 2 we define our semimartingaleenvironment, formulate our assumptions on the pricing kernel, and define zero-coupon bondsand forward measures. In Section 3 we briefly recall the risk-neutral factorization. In Section

3

4 we formulate our central result, Theorem 4.1, that gives a sufficient condition for existenceof the long bond and the long forward measure. In Section 5 we present extensions of Hansenand Jagannathan (1991), Alvarez and Jermann (2005) and Bakshi and Chabi-Yo (2012) boundsto our general semimartingale environment. In Section 6 we apply Theorem 4.1 to explicitlyobtain the long-term factorization in non-Markovian Heath-Jarrow-Morton models. In Section 7we apply Theorem 4.1 to explicitly obtain the long-term factorization in non-Markovian modelof Brody and Hughston (2013) of social discount rates. Section 8 studies the Markovian en-vironment. Section 9 gives an economic interpretation of Ross’ recovery without the Markovproperty. Proofs and supporting technical results are collected in appendices.

2 Semimartingale Pricing Kernels and Forward Measures QT

We work on a complete filtered probability space (Ω,F , (Ft)t≥0,P) with the filtration (Ft)t≥0

satisfying the usual conditions of right continuity and completeness. We assume that F0 istrivial modulo P. All random variables are identified up to almost sure equivalence. We chooseall semimartingales to be right continuous with left limits (RCLL) without further mention (thereader is referred to Jacod and Shiryaev and Protter for stochastic calculus of semimartingales;a summary of some key results can also be found in Jacod and Protter ()). For a RCLL processX, X− denotes the process of its left limits, (X−)t = lims→t,s<tXt for t > 0 and (X−)0 = X0

by convention (cf. Jacod and Shiryaev (2003)). We assume absence of arbitrage and tradingfrictions so that there exists a pricing kernel (state-price density) process S = (St)t≥0 satisfyingthe following assumptions.

Assumption 2.1. (Semimartingale Pricing Kernel) The pricing kernel process S is astrictly positive semimartingale with S0 = 1, S− is strictly positive, and EP[ST

St] < ∞ for all

T > t ≥ 0.

For each pair of times t and T , 0 ≤ t ≤ T < ∞, the pricing kernel defines a family of pricingoperators (Pt,T )0≤t≤T mapping time-T payoffs Y (FT -measurable random variables) into theirtime-t prices Pt,T (Y ) (Ft-measurable random variables):

Pt,T (Y ) = EP[STY

St

∣∣∣∣Ft

], t ∈ [0, T ].

where ST /St is the stochastic discount factor from T to t. Pt,T are positive, linear, and satisfytime consistency

Ps,t(Pt,T (Y )) = Ps,T (Y )

for Y ∈ FT .In particular, a T -maturity zero-coupon bond has a unit cash flow at time T and a price

process

P Tt = Pt,T (1) = EP

[ST

St

∣∣∣∣Ft

], s ≤ t ≤ T.

Under our assumptions, for each maturity T > 0 the zero-coupon bond price process (P Tt )t∈[0,T ]

is a positive semimartingale such that P TT = 1 and the process (MT

t := StPTt /P T

0 )t∈[0,T ] is apositive martingale (our integrability assumption on the stochastic discount factor implies that

4

EP[P Tt ] < ∞ for all future times t and bond maturities T > t). For each T , we can thus write a

factorization for the pricing kernel on the time interval [0, T ]:

St =P T0

P Tt

MTt .

We can then use the martingale MTt to define a new probability measure

QT |Ft = MTt P|Ft , t ∈ [0, T ]. (2.1)

This is the T -forward measure originally discovered by Jarrow (1987) and later independently byGeman (1989) and Jamshidian (1989) (see also Geman et al. (1995)). Under QT the T -maturityzero-coupon bond serves as the numeraire, and the pricing operator reads:

Ps,t(Y ) = EQT

[P Ts

P Tt

Y

∣∣∣∣Fs

]for Y ∈ Ft and s ≤ t ≤ T .

The forward measure is defined on Ft for t ≤ T . We now extend it to Ft for all t ≥ 0as follows. Fix T and consider a self-financing roll-over strategy that starts at time zero byinvesting one unit of account in 1/P T

0 units of the T -maturity zero-coupon bond. At time T thebond matures, and the value of the strategy is 1/P T

0 units of account. We roll the proceeds overby re-investing into 1/(P T

0 P 2TT ) units of the zero-coupon bond with maturity 2T . We continue

with the roll-over strategy, at each time kT re-investing into the bond P(k+1)TkT . We denote the

wealth process of this self-financing strategy BTt :

BTt =

P(k+1)Tt∏k

i=0 P(i+1)TiT

, t ∈ [kT, (k + 1)T ) k = 0, 1, . . . .

It is clear by construction that the process StBTt extends the martingale MT

t to all t ≥ 0, and,thus, Eq.(2.1) defines the T -forward measure on Ft for all t ≥ 0, where T now has the meaningof duration of the compounding interval. Under the forward measure QT extended to all Ft,the roll-over strategy (BT

t )t≥0 with compounding interval T serves as the new numeraire. Wecontinue to call the measure extended to all Ft for t ≥ 0 the T -forward measure and use the samenotation, as it reduces to the standard definition of the forward measure on Ft for t ∈ [0, T ].

Since the roll-over strategy (BTt )t≥0 and the positive martingale MT

t = StBTt are now defined

for all t ≥ 0, we can write the T -maturity factorization of the pricing kernel:

St =1

BTt

MTt , t ≥ 0. (2.2)

We will now investigate two limits, the short-term limit T → 0 and the long-term limit T → ∞.

3 Short-term Limit T → 0, Implied Savings Account, and RiskNeutral Measure

In this Section we also assume that S is a special semimartingale, i.e. the process of finitevariation in the semimartingale decomposition of S into the sum of a local martingale and a

5

process of finite variation can be taken to be predictable (this assumption is only made in thisSection and is not made in the rest of the paper). Since both S and S− are strictly positive, Sadmits a unique factorization (cf. Jacod (1979), Proposition 6.19 or Doberlein and Schweizer(2001), Proposition 2):

St =1

AtMt, (3.1)

where A is a strictly positive predictable process of finite variation with A0 = 1 and M a strictlypositive local martingale with M0 = 1.

Assumption 3.1. (Existence of Implied Savings Account) The local martingale M in thefactorization Eq.(3.1) of the pricing kernel is a true martingale.

Under this assumption, we can define a new probability measure Q|Ft = MtP|Ft equivalentto P on each Ft. Under Q the pricing operators read:

Pt,T (Y ) = EQ[At

ATY

∣∣∣∣Ft

].

The process A is called implied savings account implied by the term structure of zero-couponbonds (P T

t )0≤t≤T (see Doberlein and Schweizer (2001) and Doberlein et al. (2000) for a definitivetreatment in the semimartingale framework, and Rutkowski (1996) and Musiela and Rutkowski(1997) for earlier references). The name implied savings account is justified since A is a pre-dictable process of finite variation. If either it is directly assumed that there is a tradeableasset with this price process, or it can be replicated by a self-financing trading strategy in othertradeable assets, then this asset or self-financing strategy is locally riskless due to the lack of amartingale component. Under either assumption, Q plays the role of the risk-neutral measurewith the implied savings account A serving as the numeraire asset.

Furthermore, Doberlein and Schweizer (2001) explicitly show that under some additionaltechnical conditions the implied savings account A can, in fact, be identified with the limit B0

of self-financing roll-over strategies BT as the roll-over interval T goes to zero, which they callthe classical savings account (their analysis is in turn based on the work of Bjork et al. (1997)who also investigate the roll-over strategy). We can then identify the risk-neutral measure Qusing the implied savings account as the numeraire with the measure Q0 using the limitingroll-over strategy as T → 0 (classical savings account) as the numeraire.

If we assume that the pricing kernel S is a supermartingale, then, by multiplicative Doob-Meyer decomposition (cf. part 2 of Proposition 2 in Doberlein and Schweizer (2001)), the impliedsavings account A is non-decreasing. It is then globally riskless (in the sense of no decline invalue). If we assume that the implied savings account A is absolutely continuous, we can write

At = e∫ t0 rsds,

where r is the short-term interest rate process (short rate). If we assume that both S is asupermartingale and the implied savings account A is absolutely continuous, then the shortrate is non-negative. In this paper, unless explicitly stated otherwise in examples, we generallyneither assume that A is non-decreasing nor that it is absolutely continuous (i.e. that theshort rate exists). In fact, in Section 4 we do not generally assume that the implied savingsaccount exists, i.e. we do not generally make Assumption 3.1 (But Assumption 3.1 togetherwith the absolute continuity of A is automatically satisfied under assumptions of Section 6 inHJM models).

6

4 The Long-term Limit T → ∞, Long Bond, and Long ForwardMeasure

The T -forward factorization (2.2) decomposes the pricing kernel into discounting at the rateof return on the T -maturity bond and a further risk adjustment. The short-term factorization(3.1) emerging in the short-term limit T → 0 as discussed in the previous section decomposesthe pricing kernel into discounting at the rate of return on the riskless asset and adjusting forrisk. In contrast, in this section our main focus is the long-term limit T → ∞.

Definition 4.1. (Long Bond) If the wealth processes (BTt )t≥0 of the roll-over strategies in

T -maturity bonds converge to a strictly positive semimartingale (B∞t )t≥0 uniformly on compacts

in probability as T → ∞, i.e. for all t > 0 and K > 0

limT→∞

P(sups≤t

|BTs −B∞

s | > K) = 0,

we call the limit the long bond.

Definition 4.2. (Long Forward Measure) If there exists a measure Q∞ locally equivalent toP such that the T -forward measures converge strongly to Q∞ on each Ft, i.e.

limT→∞

QT (A) = Q∞(A)

for each A ∈ Ft and each t ≥ 0, we call the limit the long forward measure and denote it L.

The following theorem is the central result of this paper. It gives an explicit sufficientcondition easy to verify in applications that ensures stronger modes of convergence (Emery’ssemimartingale convergence to the long bond and convergence in total variation to the longforward measure).

Theorem 4.1. (Long-Term Factorization of the Pricing Kernel) Suppose that for eacht > 0 the sequence of strictly positive random variables MT

t converge in L1 to a strictly positivelimit:

limT→∞

EP[|MTt −M∞

t |] = 0 (4.1)

with M∞t > 0 a.s. Then the following results hold.

(i) (M∞t )t≥0 is a positive P-martingale and (MT

t )t≥0 converge to (M∞t )t≥0 in the semimartingale

topology.(ii) Long bond (B∞

t )t≥0 exists and (BTt )t≥0 converge to (B∞

t )t≥0 in the semimartingale topology.(iii) The pricing kernel possesses a factorization

St =1

B∞t

M∞t . (4.2)

(iv) T -forward measures QT converge to the long forward measure L in total variation on eachFt, and

L|Ft = M∞t P|Ft

for each t ≥ 0.

7

The proof of Theorem 4.1 is given in Appendix A. The value of the long bond at time t, B∞t ,

has the interpretation of the gross return earned starting from time zero up to time t on holdingthe zero-coupon bond of asymptotically long maturity. The long-term factorization of the pricingkernel (4.2) into discounting at the rate of return on the long bond and a martingale componentencoding a further risk adjustment extends the long-term factorization of Alvarez and Jermann(2005) and Hansen and Scheinkman (2009) to general semimartingale environments.

Remark 4.1. By Theorem II.5 in Memin (1980), Emery’s semimartingale convergence is in-variant under locally equivalent measure transformations. Furthermore, Eq.(4.1) can be writtenunder any locally equivalent probability measure. Namely, let Vt be a strictly positive semimartin-gale with V0 = 1 and such that StVt is a martingale, and define QV |Ft = StVtP|Ft for all t ≥ 0.Then Eq.(4.1) reads under QV :

limT→∞

EQV[|B

Tt

Vt− B∞

t

Vt|] = 0. (4.3)

In applications it helps us choose a convenient measure QV to verify this condition in a concretemodel.

We now consider a special class of pricing kernels.

Definition 4.3. We say that an economy is long-term risk-neutral if the condition in Theorem4.1 holds and the pricing kernel has the form

St =1

B∞t

for all t ≥ 0.

By definition, in a long-term risk-neutral economy M∞t = 1, P = L and we immediately have

Proposition 4.1. In a long-term risk-neutral economy the long bond is growth optimal (i.e. ithas the highest expected log return).

Proof. Consider the value process Xt of an asset or a self-financing trading strategy with X0 = 1and such that StXt = Xt/B

∞t is a P martingale. Thus, E[Xt/B

∞t ] = 1. By Jensen’s inequality,

E[log(Xt/B∞t )] ≤ logE[Xt/B

∞t ] = 0, which implies E[log(Xt)] ≤ E[log(B∞

t )], i.e., the long bondhas the highest expected log return.

5 Pricing Kernel Bounds

We now formulate Hansen and Jagannathan (1991), Alvarez and Jermann (2005) and Bakshiand Chabi-Yo (2012) bounds in our general semimartingale environment. The celebrated Hansenand Jagannathan (1991) theorem states that the ratio of the standard deviation of a stochasticdiscount factor to its mean exceeds the Sharpe ratio attained by any portfolio. This result holdsin our general semimartingale framework. For any T > t ≥ 0 and an FT -measurable randomvariable X, let σt(X) :=

√Et[X2]− (Et[X])2 and σ(X) :=

√E[X2]− (E[X])2 denote its condi-

tional and unconditional standard deviation. To formulate Alvarez and Jermann (2005) bounds,let Lt(X) := logEt[X]− Et[logX] and L(X) := logE[X]− E[logX] denote the conditional andunconditional Thiel’s second entropy measure, respectively, where Et[·] = EP[·|Ft].

8

Theorem 5.1. (Hansen-Jagannathan Bounds) Let Vt be a semimartingale such that StVt

is a martingale. Then the following inequalities hold.

(i) σ(ST

St

)/E[ST

St

]≥

∣∣∣E[VT

Vt

]− 1

E[P Tt ]

∣∣∣/σ(VT

Vt

).

(ii) σt

(ST

St

)/Et[

ST

St] ≥

∣∣∣Et

[VT

Vt

]− 1

P Tt

∣∣∣/σt

(VT

Vt

).

The proof is based only on the martingality of StVt and Holder’s inequality and is givenin Appendix B. If σ(ST /St) = ∞ and/or σ(VT /Vt) = ∞, the bound holds trivially. Whent = 0, the right-hand side of (i) is the Sharpe ratio of a self-financing portfolio with the wealthprocess V (ratio of the excess return on the portfolio over the risk-free return on investing in theT -maturity zero-coupon bond relative to the standard deviation of the portfolio). When t > 0,it can be interpreted as the forward Sharpe ratio over the future time interval [t, T ] as seen fromtime zero. The right hand side of (ii) can be interpreted as the conditional Sharpe ratio at timet. The economic power of this inequalities is that that they apply to all self-financing portfolioswith wealth processes V such that StVt are martingales.

The next result extends Bakshi and Chabi-Yo (2012) bounds on the factors in the long-termfactorization. This result can be viewed as an elaboration on Hansen-Jagannathan bounds forthose pricing kernels that admit the long-term factorization, yielding Hansen-Jagannathan typebounds for the two factors in the long-term factorizatio, as well their ratio.

Theorem 5.2. (Bakshi-Chabi-Yo Bounds) Suppose the pricing kernel admits a long-termfactorization Eq.(4.2) with the positive martingale M∞. Let Vt be a semimartingale such thatStVt is a martingale. Then the following inequalities hold.

(i) σ(M∞

T

M∞t

)≥

∣∣∣E[ VT /Vt

B∞T /B∞

t

]− 1

∣∣∣/σ( VT /Vt

B∞T /B∞

t

),

(ii) σ(B∞

t

B∞T

)≥

(E[B∞

t

B∞T

]E[B∞

T

B∞t

]− 1

)/σ(B∞

T

B∞t

),

(iii) σ(M∞

T B∞T

M∞t B∞

t

)≥

∣∣∣E[M∞T B∞

T

M∞t B∞

t

]E[ VT /Vt

(B∞T /B∞

t )2

]− 1

∣∣∣/σ( VT /Vt

(B∞T /B∞

t )2

),

(iv) σt

(M∞T

M∞t

)≥

∣∣∣Et

[ VT /Vt

B∞T /B∞

t

]− 1

∣∣∣/σt

( VT /Vt

B∞T /B∞

t

),

(v) σt

(B∞t

B∞T

)≥

(Et

[ 1

B∞T

]Et

[B∞

T

]− 1

)/σt

(B∞T

B∞t

),

(vi) σt

(M∞T B∞

T

M∞t B∞

t

)≥

∣∣∣Et

[M∞T B∞

T

M∞t B∞

t

]Et

[ VT /Vt

(B∞T /B∞

t )2

]− 1

∣∣∣/σt

( VT /Vt

(B∞T /B∞

t )2

),

The proof is essentially the same as in for the Hansen-Jagannathan bounds, is based onlyon the martingale property and Holder’s inequality, and is given in Appendix B. If eithervariance is infinite, the bounds hold trivially. Bakshi and Chabi-Yo (2012) estimate their boundsfor the volatility of the permanent (martingale) and transient components in the long-termfactorization from empirical data on returns on multiple asset classes. Just as the originalHansen-Jagannathan bounds, the economic power of these bounds is their applicability to allself-financing portfolios in an arbitrage-free model such that StVt is a martingale.

Our next result extends Alvarez and Jermann (2005) bounds on the martingale componentin the long-term factorization.

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Theorem 5.3. (Alvarez-Jermann Bounds) Suppose the pricing kernel admits a long-termfactorization Eq.(4.2) with the positive martingale M∞

t . Let Vt be a semimartingale such thatStVt is a martingale. Then the following inequalities hold.

(i) L(M∞

T

M∞t

) ≥ E[logVTB

∞t

VtB∞T

],

(ii) Lt(M∞

T

M∞t

) ≥ Et[logVTB

∞t

VtB∞T

].

(iii) If E[log

PTt VT

Vt

]+ L(P T

t ) > 0, thenL(M∞

TM∞

t

)L(STSt

) ≥ min

1,E[log

VTB∞t

VtB∞T

]E[log

PTt VT

Vt

]+ L(P T

t )

.

(iv) If Et[logPTt VT

Vt] > 0, then

Lt

(M∞

TM∞

t

)Lt

(STSt

) ≥ min

1,Et

[log

VTB∞t

VtB∞T

]Et

[log

PTt VT

Vt

] .

The proof is based on the martingale property and Jensen’s inequality and is given in Ap-pendix B. The right hand side of the Alvarez-Jermann bound is the expected excess log-returnover the time period [t, T ] earned on the self-financing portfolio with the wealth process Vrelative to the return earned from holding the long bond B∞ over this time period.

6 Long-Term Factorization in Heath-Jarrow-Morton Models

The classical Heath et al. (1992) framework assumes that the family of zero-coupon bond pro-cesses (P T

t )t∈[0,T ], T ≥ 0 is sufficiently smooth across the maturity parameter T so that thereexists a family of instantaneous forward rate processes (f(t, T ))t∈[0,T ], T ≥ 0 such that

P Tt = e−

∫ Tt f(t,s)ds,

and for each T the forward rate is assumed to follow an Ito process

df(t, T ) = µ(t, T )dt+ σ(t, T ) · dW Pt , (6.1)

driven by an n-dimensional Brownian motion W P = (W P,jt )t≥0, j = 1, . . . , n. For each fixed

maturity date T , the volatility of the forward rate (σ(t, T ))t∈[0,T ] is an n-dimensional Ito process.In (6.1) · denotes the Euclidean dot product in Rn. Heath et al. (1992) show that absence ofarbitrage implies the drift condition

µ(t, T ) = (−γ(t) + σP (t, T )) · σ(t, T ),

where γ(t) is the (negative of the) d-dimensional market price of Brownian riskW P and σP (t, T ) :=∫ Tt σ(t, u)du. By Ito formula, zero-coupon bond prices follow Ito processes given by

dP (t, T )

P (t, T )= (r(t) + γ(t) · σP (t, T ))dt− σP (t, T ) · dW P

t ,

where r(t) = f(t, t) is the short rate. Thus, σP (t, T ) defined early as the integral of the forwardrate volatility σ(t, u) in maturity date u from t to T is identified with the zero-coupon bondvolatility. The classical HJM framework treats the forward rate of each maturity as an Ito

10

process. We thus have a family of Ito processes parameterized by the maturity date. Heathet al. (1992) give sufficient conditions on the forward rate volatility and the market price of riskfor the model to be well defined. Jin and Glasserman (2001) show how HJM models can besupported in a general equilibrium.

An alternative point of view is to treat the forward curve ft as a single process takingvalues in an appropriate function space of forward curves. To this end, the Musiela (1993)parameterization ft(x) := f(t, t + x) of the forward curve is convenient. Now we work withthe process (ft)t≥0 taking values in an appropriate space of functions on R+ (x ∈ R+), wherex denotes time to maturity (so that t + x is the maturity date). Furthermore, this point ofview immediately extends the HJM framework to allow the driving Brownian motion W to beinfinite-dimensional (see Prato and Zabczyk (2014) for mathematical foundations of infinite-dimensional stochastic analysis). Here we follow Filipovic (2001), who gives a comprehensivetreatment of this point of view. The forward curve dynamics now reads:

dft = (Aft + µt)dt+ σt · dW Pt . (6.2)

The infinite-dimensional standard Brownian motion W P = (W P,jt )t≥0, j = 1, 2, . . . is a se-

quence of independent standard Browian motions adapted to (Ft)t≥0 (see Filipovic (2001)

Chapter 2). We continue to use notation σt · dW Pt =

∑j∈N σj

t dWP,jt in the infinite-dimensional

case (the finite-dimensional case arises by setting σjt ≡ 0 for all j > n for some n). The forward

curve (ft)t≥0 is now a process taking values in the Hilbert space Hw that we will define shortly.

The drift µt = µ(t, ω, ft) and volatility σjt = σj(t, ω, ft) take values in the same Hilbert space

Hw and depend on ω and ft. To lighten notation we often do not show this dependence explic-itly. The additional term Aft in the drift in Eq.(6.2) arises from Musiela’s parameterization,where the operator A is interpreted as the first derivative with respect to time to maturity,Aft(x) = ∂xft(x) and is defined more precisely below as the operator in Hw.

Following Filipovic (2001), we next define the Hilbert space Hw of forward curves and giveconditions on the volatility and drift to ensure that the solution of the HJM evolution equationEq.(6.2) exists in the appropriate sense (the forward curve stays in its prescribed space Hw asit evolves in time) and specifies an arbitrage-free term structure. Let w : R+ → [1,∞) be anon-decreasing C1 function such that

∫∞0

dxw1/3(x)

< ∞. We define

Hw := h ∈ L1loc(R+) | ∃h′ ∈ L1

loc(R+) and ∥h∥w < ∞,

where

∥h∥2w := |h(0)|2 +∫R+

|h′(x)|2w(x)dx.

Hw is defined as the space of locally integrable functions, with locally integrable (weak) deriva-tives, and with the finite norm ∥h∥w. The finiteness of this norm imposes tail decay on thederivative h′ of the function (forward curve) in time to maturity such that it decays to zero astime to maturity tends to infinity fast enough so the derivative is square integrable with theweight function w, which is assumed to grow fast enough so that the integral

∫∞0

dxw1/3(x)

is finite.

By Holder’s inequality, for all h ∈ Hw∫R+

|h′(x)|dx ≤(∫

R+

|h′(x)|2w(x)dx)1/2(∫

R+

w−1(x)dx)1/2

< ∞.

11

Thus, h(x) converges to the limit h(∞) ∈ R as x → ∞, which can be interpreted as thelong forward rate. In other words, all forward curves in Hw flatten out at asymptotically longmaturities. This is an instance of the theorem due to Dybvig et al. (1996) (see Brody andHughston (2013) for a recent extension, proof, and related references). We note that existenceof the long forward rate is not in itself sufficient for existence of the long bond. The sufficientcondition will be given below. The space Hw equipped with ∥h∥w is a separable Hilbert space.Define a semigroup of translation operators on Hw by (Ttf)(x) = f(t+ x). By Filipovic (2001)Theorem 5.1.1, it is strongly continuous in Hw, and we denote its infinitesimal generator by A.This is the operator that appears in the drift in Eq.(6.2) due to the Musiela re-parameterization.(The reader can continue to think of it as the derivative of the forward rate in time to maturity.)

We next give conditions on the drift and volatility. First we need to introduce some notations.Define the subspace H0

w ⊂ Hw by

H0w = f ∈ Hw such that f(∞) = 0.

For any continuous function f on R+, define the continuous function Sf : R+ → R by

(Sf)(x) := f(x)

∫ x

0f(η)dη, x ∈ R+. (6.3)

This operator is used to conveniently express the celebrated HJM arbitrage-free drift condition.By Filipovic (2001) Theorem 5.1.1, there exists a constant K such that ∥Sh∥w ≤ K∥h∥2w for allh ∈ Hw with h ∈ H0

w. Local Lipschitz property of S is proved in Filipovic (2001) Corollary 5.1.2,which is used to ensure existence and uniqueness of solution to the HJM equation. Namely, Smaps H0

w to H0w and is locally Lipschitz continuous:

∥Sg − Sh∥w ≤ C(∥g∥w + ∥h∥w)∥g − h∥w,∀g, h ∈ H0w,

where the constant C only depends on w.Consider ℓ2, the Hilbert space of square-summable sequences, ℓ2 = v = (vj)j∈N ∈ RN|∥v∥2ℓ2 :=∑

j∈N |vj |2 < ∞. Let ej denote the standard orthonormal basis in ℓ2. For a separable Hilbert

space H, let L02(H) denote the space of Hilbert-Schmidt operators from ℓ2 to H with the Hilbert-

Schmidt norm∥ϕ∥2L0

2(H) :=∑j∈N

∥ϕj∥2H < ∞,

where ϕj := ϕej . We shall identify the operator ϕ with its H-valued coefficients (ϕj)j∈N.We are now ready to give conditions on the market price of risk and volatility. Recall that

we have a filtered probability space (Ω,F , (Ft)t≥0,P). Let P be the predictable sigma-field.For any metric space G, we denote by B(G) the Borel sigma-field of G.

Assumption 6.1. (Conditions on the Parameters and the Initial Forward Curve)(i) The initial forward curve f0 ∈ Hw.(ii) The (negative of the) market price of risk γ is a measurable function from (R+×Ω×Hw,P⊗B(Hw)) into (ℓ2,B(ℓ2)) such that there exists a function Γ ∈ L2(R+) that satisfies

∥γ(t, ω, h)∥ℓ2 ≤ Γ(t) for all (t, ω, h). (6.4)

(iii) The volatility σ = (σj)j∈N is a measurable function from (R+ × Ω×Hw,P ⊗ B(Hw)) into(L0

2(H0w),B(L0

2(H0w))). It is is assumed to be Lipschitz continuous in h and uniformly bounded,

i.e. there exist constants D1, D2 such that for all (t, ω) ∈ R+ × Ω and h, h1, h2 ∈ Hω

∥σ(t, ω, h1)− σ(t, ω, h2)∥L02(Hw) ≤ D1∥h1 − h2∥Hw , ∥σ(t, ω, h)∥L0

2(Hw) ≤ D2. (6.5)

12

In the case when W P is finite-dimensional, simply replace ℓ2 with Rn. Filipovic (2001)Remark 2.4.3 states that the Lipschitz constant D1 may depend on T , i.e. (6.5) holds for all(t, ω) ∈ [0, T ]× Ω.

The drift µt = µ(t, ω, ft) in (6.2) is defined by the HJM drift condition:

µ(t, ω, ft) = αHJM(t, ω, ft)− (γ · σ)(t, ω, ft),

where αHJM(t, ω, ft) =∑

j∈N Sσj(t, ω, ft), where S is the previously defined operator (6.3).The following theorem summarizes the properties of the HJM model (6.2) in this setting.

Theorem 6.1. (HJM Model) (i) Eq.(6.2) has a unique continuous weak solution.(ii) The pricing kernel has the risk-neutral factorization

St =1

AtMt

with the implied savings account A given by

At = exp( ∫ t

0fs(0)ds

)and the martingale

Mt = exp(− 1

2

∫ t

0∥γs∥2ℓ2ds+

∫ t

0γs · dW P

s

)defining the risk-neutral measure Q|Ft = MtP|Ft. The process

WQt := W P

t −∫ t

0γsds

is an (infinite-dimensional) standard Brownian motion under Q.(iii) The T -maturity bond price P T

t follows the process:

P Tt

P T0

= At exp(∫ t

0γs · σT

s ds−∫ t

0σTs · dW P

s − 1

2

∫ t

0∥σT

s ∥ℓ2ds),

where the volatility of the T -maturity bond is

σTt =

∫ T−t

0σt(u)du.

For the definition of a weak solution used here see Filipovic (2001) Definition 2.4.1. Theproof of Theorem 6.1 follows from the results in Filipovic (2001) and summarized for the readers’convenience in Appendix C.

We are now ready to formulate the main result of this section.

Theorem 6.2. (Long-term Factorization in the HJM Model) Suppose the initial forwardcurve and the market price of risk satisfy Assumption 6.1 (i) and (ii). Suppose the volatilityσ = (σj)j∈N is a measurable function from (R+×Ω×Hw,P⊗B(Hw)) into (L0

2(H0w′),B(L0

2(H0w′)))

and is Lipschitz continuous in h and uniformly bounded as in 6.1 (ii), where H0w′ ⊆ H0

w with w′

satisfying the estimate1

w′(x)= O(x−(3+ϵ)). (6.6)

13

Then the following results hold.(i) Eq.(4.1) in Theorem 4.1 holds in the HJM model and, hence, all results in Theorem 4.1 hold.(ii) Define

σ∞t :=

∫ ∞

0σt(u)du. (6.7)

The long bond follows the process:

B∞t = At exp

( ∫ t

0γs · σ∞

s ds−∫ t

0σ∞s · dW P

s − 1

2

∫ t

0∥σ∞

s ∥2ℓ2ds)

(6.8)

with volatility σ∞t . The pricing kernel admits the long-term factorization:

St =1

B∞t

M∞t

with the martingale

M∞t = exp

(− 1

2

∫ t

0∥γs − σ∞

s ∥2ℓ2ds+∫ t

0(γs − σ∞

s ) · dW Ps

)(6.9)

The long forward measure L is given by dLdP |Ft = M∞

t .(iii) Under L

WLt := W P

t +

∫ t

0(σ∞

s − γs)ds

is an (infinite-dimensional) standard Brownian motion, and the L-dynamics of the forward curveis:

dft = (Aft + αHJMt − σ∞

t · σt)dt+ σt · dWLt .

The proof is given in Appendix C. The sufficient condition on the forward curve volatility toensure existence of the long bond is a slight strengthening of Filipovic’s condition in Assumption6.1. We require that the weight function w′ in the weighted Sobolev space Hw′ where thevolatility components take their values satisfies the estimate (6.6), a slight strenghening ofFilipovic’s assumption on the weight w of Hw where the forward curves evolve. Note that wedo not impose this estimate on w in the space of forward curves Hw, only on the space Hw′

featured in the definition of the forward curve volatility. We note that our sufficient conditionsensure that the volatility of the long bond (6.7) is well defined (see proof in Appendix C).

Theorem 6.2 provides a fully explicit long-term factorization in the generally non-MarkovianHJM model driven by an infinite-dimensional Brownian motion and illustrates the power of ourgeneral framework.

Example 6.1. Gaussian HJM modelsWhen the forward curve volatility σ is deterministic, the model reduces to the Gaussian HJMmodel. In this case the conditions on σ in Theorem 6.2 simplify to requiring that σ(t, ω, h) =σ(t) ∈ L0

2(H0w′) and is uniformly bounded for some w′ such that H0

w′ ⊆ H0w and 1/w′(x) =

O(x−(3+ϵ)).When the volatility σ is also independent of time, the model reduces to the generalized Vasicek

model extending the classical one-dimensional Vasicek model with σ1 = σre−κx and σj = 0 for

j ≥ 2. When κ > 0, the short rate rt = ft(0) follows a mean-reverting Vasicek / OU process withconstant volatility σr and the rate of mean reversion κ. Our conditions on σ are automaticallysatisfied in this case. In particular, in this case the volatility of the long bond is constant,σ∞t = σr/κ.

14

7 Long-Term Factorization in Models of Social Discounting

Our next example is a non-Markovian model of social discount rates due to Brody and Hughston(2013). These authors put forward an arbitrage-free model of social discount rates that is fullyconsistent with the semimartingale approach to pricing kernels and, at the same time, featuresthe term structure of zero-coupon bond prices (discount factors) P T

t that decays as O(T−1)(or as more general negative power) as maturity becomes asymptotically long. This avoids theheavy discounting of the welfare of future generations in the distant future present in models withexponentially decaying term structures. The problem of determining appropriate social discountrates is of importance in evaluating policies to combat climate change (see Arrow (1995), Arrowet al. (1996), Weitzman (2001) for discussions). Here we demonstrate that Brody and Hughston(2013) models of social discount rates verify our condition (4.1) and, thus, our Theorem 4.1holds in this class of models. Moreover, we explicitly construct the long-term factorization inthese models.

Let a(t) and b(t) be two deterministic strictly positive continuous function of time suchthat limt→∞ ta(t) = a and limt→∞ tb(t) = b for some non-negative constants a and b such thata + b > 0 and normalized so that a(0) + b(0) = 1 (for our purposes we slightly strengthen theassumptions of Brody and Hughston (2013) by assuming that the limits exists, instead of thelimits inferior). Let (Mt)t≥0 be a positive P-martingale with M0 = 1 and define the pricingkernel by

St = a(t) + b(t)Mt.

Then it is immediate that

P Tt =

a(T ) + b(T )Mt

a(t) + b(t)Mt. (7.1)

From the assumptions on a(t) and b(t) we have

limT→∞

(TP Tt ) =

a+ bMt

a(t) + b(t)Mt

for each fixed t ≥ 0, which implies that P Tt is O(T−1) and, thus, the continuously compounded

exponential long rate defined as − limT→∞1T lnP T

t vanishes in this model. Nevertheless, theso-called tail-Pareto long rates are finite and non-vanishing (see Brody and Hughston (2013)).In particular, for each t ≥ 0

limT→∞

BTt = lim

T→∞

P Tt

P T0

=a+ bMt

(a+ b)(a(t) + b(t)Mt)a.s.

We now verify that MTt = StB

Tt satisfy Eq.(4.1) with

M∞t =

a+ bMt

a+ b

so that

B∞t = M∞

t /St =a+ bMt

(a+ b)(a(t) + b(t)Mt).

By direct calculation, we have∣∣∣StP Tt

P T0

− StB∞t

∣∣∣ = |(ab(T )− a(T )b)(Mt − 1)|(a(T ) + b(T ))(a+ b)

≤ |aTb(T )− Ta(T )b|(Ta(T ) + Tb(T ))(a+ b)

(Mt + 1).

15

Since limT→∞(aTb(T )−Ta(T )b) = 0, limT→∞(Ta(T )+Tb(T )) = a+ b and Mt+1 is integrableunder P, it is clear that

limT→∞

EP[∣∣∣St

P Tt

P T0

− StB∞t

∣∣∣] = 0.

Thus, (4.1) is satisfied and Theorem 4.1 holds.We can apply Girsanov theorem for semimartingales to explicitly obtain dynamics of M

under the the long forward measure L defined by the martingale M∞. In particular, assumethat the (square brackets) quadratic variation [M ] of M is P-locally integrable. Then we candefine the predictable (sharp brackets) quadratic variation ⟨M⟩ so that [M ] − ⟨M⟩ is a P-localmartingale. Then by Girsanov Theorem (cf. Jacod and Shiryaev (2003) p.168 Theorem 3.11)the P-martingale becomes L-semimartingale with the canonical decomposition

Mt = Mt +

∫ t

0

bd⟨M⟩sa+ bMs−

,

where Mt = Mt −∫ t0

bd⟨M⟩sa+bMs−

is a L-local martingale and∫ t0

bd⟨M⟩sa+bMs−

is a predictable process

of finite variation (“drift”). This example can also be straightforwardly generalized to pricingkernels of the form St = a(t) + b(t)Mt + c(t)Nt driven by several martingales (see Brody andHughston (2013)).

Example 7.1. Rational Lognormal Model of Social Discount RatesWe now specify the martingale M to be the Dooleans-Dade exponential martingale of a P-Brownian motion, Mt = eσW

Pt −

12σ2t. The corresponding pricing kernel St = a(t)+b(t)eσW

Pt −

12σ2t

defines the Flesakerand and Hughston (1996) term structure model (7.1). In this case [M ]t =⟨M⟩t =

∫ t0 σ

2M2s ds and

Mt = Mt −∫ t

0

bσ2M2s

a+ bMsds

is a L-local martingale.We can give a more detailed characterization by Girsanov’s theorem. Since L|Ft =

a+bMta+b P|Ft,

WLt = W P

t −∫ t

0

bσMs

a+ bMsds

is L-standard Brownian motion. Thus M satisfies the SDE

dMt =bσ2M2

t

a+ bMtdt+ σMtdW

Lt ,

under L, and dMt = σMtdWLt . Since EL[

∫ t0 σ

2M2s ds] =

∫ t0 E

P[σ2M2s

(a+bMsa+b

)]ds < ∞, Mt is a

true L-martingale.

8 Long-Term Factorization in Markovian Economies

8.1 Theory

We now further specify our model by assuming that the underlying filtration is generated bya Markov process X and the pricing kernel is a positive multiplicative functional of X. More

16

precisely, the stochastic driver of all economic uncertainty is a conservative Borel right process(BRP) X = (Ω,F , (Ft)t≥0, (Xt)t≥0, (Px)x∈E). A BRP is a continuous-time, time-homogeneousMarkov process taking values in a Borel subset E of some metric space (so that E is equippedwith a Borel sigma-algebra E ; in applications we can think of E as a Borel subset of theEuclidean space Rd), having right-continuous paths and possessing the strong Markov property(i.e., the Markov property extended to stopping times). The probability measure Px governs thebehavior of the process (Xt)t≥0 when started from x ∈ E at time zero. If the process starts froma probability distribution µ, the corresponding measure is denoted Pµ. A statement concerningω ∈ Ω is said to hold P-almost surely if it is true Px-almost surely for all x ∈ E. The informationfiltration (Ft)t≥0 in our model is the filtration generated by X completed with Pµ-null sets forall initial distributions µ of X0. It is right continuous and, thus, satisfies the usual hypothesisof stochastic calculus. X is assumed to be conservative, i.e. Px(Xt ∈ E) = 1 for each initialx ∈ E and all t ≥ 0 (the process does not exit the state space E in finite time, i.e. no killing orexplosion).

Cinlar et al. (1980) show that stochastic calculus of semimartingales defined over a rightprocess can be set up so that all key properties hold simultaneously for all starting points x ∈ Eand, in fact, for all initial distributions µ of X0. In particular, an (Ft)t≥0-adapted process S isan Px-semimartingale (local martingale, martingale) simultaneously for all x ∈ E and, in fact,for all Pµ, where µ is the initial distribution (of X0). With some abuse of notation, in this sectionwe simply write P where, in fact, we are dealing with the family of measures (Px)x∈E indexed bythe initial state x. Correspondingly, we simply say that a process is a P-semimartingale (localmartingale, martingale), meaning that it is a Px-semimartingale (local martingale, martingale)for each x ∈ E.

The advantage of working in this generality of Borel right processes is that we can treatprocesses with discrete state spaces (Markov chains), diffusions in the whole Euclidean space orin a domain with a boundary and some boundary behavior, as well as pure jump and jump-diffusion processes in the whole Euclidean space or in a domain with a boundary, all in a unifiedfashion.

We assume that the pricing kernel (St)t≥0 is a positive ((Ft)t≥0,P)-semimartingale that, inaddition to Assumption 2.1, is also a multiplicative functional of X, i.e.

St+s(ω) = St(ω)Ss(θt(ω)),

where θt : Ω → Ω is the shift operator (i.e. Xt(θs(ω)) = Xt+s(ω)). We also assume integrabilityEPx[St] < ∞ for all t and each x ∈ E so that Assumption 2.1 is satisfied.Then the time-s price of a payoff f(Xt) at time t ≥ s ≥ 0 is

Pt−sf(Xs) = EP[St

Ssf(Xt)

∣∣∣∣Fs

].

where we used the Markov property and time homogeneity and introduced a family of pricingoperators (Pt)t≥0:

Ptf(x) := EPx[Stf(Xt)],

where EPx denotes the expectation with respect to Px. The pricing operator Pt maps the payoff

function f at time t into its present value function at time zero as a function of the initial stateX0 = x.

17

Suppose Pt have a positive eigenfunction π satisfying

Ptπ(x) = e−λtπ(x)

for some λ ∈ R and all t > 0, x ∈ E. Then the process

Mπt = Ste

λt π(Xt)

π(X0)

is a positive P-martingale. The key observation of Hansen and Scheinkman (2009) is that thenthe pricing kernel admits a factorization

St = e−λtπ(X0)

π(Xt)Mπ

t . (8.1)

Since Mπt is a positive P-martingale starting from one, we can define a new probability measure

Qπ (eigen-measure) associated with the eigenfunction π by:

Qπ|Ft = Mπt P|Ft .

The pricing operator can then be expressed under Qπ as

Ptf(x) = e−λtπ(x)EQπ

x

[f(Xt)

π(Xt)

].

We refer the reader to Hansen and Scheinkman (2009) and Qin and Linetsky (2014) for moreon factorizations of multiplicative pricing kernels.

In general, for a given pricing kernel S, there may be no, one, or multiple positive eigen-functions. However, Qin and Linetsky (2014) prove that there exists at most one positiveeigenfunction such that the process is recurrent under Qπ (for the precise definition of a recur-rent BRP, see Qin and Linetsky (2014) Definition 2.2). If such an eigenfunction exists, Qin andLinetsky (2014) call it a recurrent eigenfunction, denote it by πR, and call the associated QπR

recurrent eigen-measure. The corresponding eigenvalue is denoted as λR. Existence of a recur-rent eigenfunction is investigated in Qin and Linetsky (2014), where several sufficient conditionsare obtained and analytical solutions are provided for a range of specific models.

We now give a sufficient condition that leads to the identification QπR = L.

Assumption 8.1. (Exponential Ergodicity) Assume a recurrent eigenfunction πR existsand under the recurrent eigen-measure QπR X satisfies the following exponential ergodicity as-sumption. There exists a probability measure ς on E and some positive constants c, α, t0 suchthat the following exponential ergodicity estimate holds for all Borel functions satisfying |f | ≤ 1:∣∣∣∣EQπR

x

[f(Xt)

πR(Xt)

]−

∫E

f(y)

πR(y)ς(dy)

∣∣∣∣ ≤ c

πR(x)e−αt (8.2)

for all t ≥ t0 and each x ∈ E.

Under this assumption, the distribution of X converges to the limiting distribution ς at theexponential rate α. Sufficient conditions for (8.2) for Borel right processes can be found inTheorem 6.1 of Meyn and Tweedie (1993).

18

Theorem 8.1. (Identification of L and QπR) If Assumption 8.1 is satisfied, then Eq.(4.1)holds with M∞

t = MπRt and, thus, Theorem 4.1 applies, the long bond is given by

B∞t = eλRt πR(Xt)

πR(X0),

and the recurrent eigen-measure coincides with the long forward measure:

QπR = L.

The proof is given in Appendix D. This result supplies a sufficient condition for existence ofthe long forward measure and the long-term factorization and is close to the results in Hansenand Scheinkman (2009), but is distinct from them in several respects. Here our developmentstarts with defining the long bond and the long forward measure. There is no issue of uniquenessby definition, but there is an issue of existence. In Markovian economies, we show that if thepricing kernel possesses a recurrent eigenfunction πR and if the Markov process moreover satisfiesthe exponential ergodicity assumption (8.1) under the recurrent eigen-measure QπR , then thelong forward measure exists and is identified with the recurrent eigen-measure. The pricingkernel then possesses the long-term factorization, and the long-term factorization is identifiedwith the eigen-factorization of Hansen and Scheinkman (2009).

We stress that Assumption 8.1 is merely a sufficient condition for the identification QπR = L.This identification may hold in models not satisfying Assumption 8.1. On the other hand, ingeneral (when Assumption 8.1 is not satisfied), it may be the case that L exists while QπR doesnot (this case is illustrated in Section 8.2.4 below), that QπR exists while L does not, or eventhat both L and QπR exist but are distinct.

8.2 Markovian Examples

8.2.1 Hunt Processes with Duals

The example in this section closely follows the class of pricing kernels investigated in Section 5.1of Qin and Linetsky (2014). In this example we assume that X is a conservative Hunt process ona locally compact separable metric space E. This entails making additional assumptions that theBorel right process X on E also has sample paths with left limits and is quasi-left continuous (nojumps at predictable stopping times, and fixed times in particular). In this section we furtherassume that the pricing kernel admits an absolutely continuous non-decreasing implied savings

account (recall Section 3) At = e∫ t0 r(Xs)ds with the non-negative short rate function r(x). The

risk-neutral measure Q can then be defined, and the pricing operators take the form

Ptf(x) = EQx [e

−∫ t0 r(Xs)dsf(Xt)]

under Q. Let Xr denote X killed at the rate r (i.e. the process is killed (sent to an isolatedcemetery state) at the first time the positive continuous additive functional

∫ t0 r(Xs)ds exceeds

an independent unit-mean exponential random variable). It is a Borel standard process (seeDefinition A.1.23 and Theorem A.1.24 in Chen and Fukushima (2011)) since it shares the samplepath with the Hunt process X prior to the killing time. The pricing semigroup (Pt)t≥0 is thenidentified with the transition semigroup of the Borel standard process Xr.

19

We further assume that there is a positive sigma-finite reference measure m with full supporton E such that Xr has a dual with respect to m. That is, there is a strong Markov process Xr

on E with the semigroup (Pt)t≥0 such that for any t > 0 and non-negative functions f and g:∫Ef(x)Ptg(x)m(dx) =

∫Eg(x)Ptf(x)m(dx).

We further make the following assumptions.

Assumption 8.2. (i) There exists a family of continuous and strictly positive functions p(t, ·, ·)on E × E such that for any (t, x) ∈ (0,∞)× E and any non-negative function f on E,

Ptf(x) =

∫Ep(t, x, y)f(y)m(dy), Ptf(x) =

∫Ep(t, y, x)f(y)m(dy).

(ii) The density satisfies: ∫E

∫Ep2(t, x, y)m(dx)m(dy) < ∞, ∀t > 0.

(iii) There exists some T > 0 such that

supx∈E

∫Ep2(t, x, y)m(dy) < ∞, sup

x∈E

∫Ep2(t, y, x)m(dy) < ∞, ∀t ≥ T.

Under these assumptions, Qin and Linetsky (2014) prove the following results.

Theorem 8.2. Suppose Assumption 8.2 is satisfied. Then the following results hold.(i) The pricing operator Pt and the dual operator Pt possess unique positive, continuous, bound-ed eigenfunctions πR(x) and πR(x) belonging to L2(E,m):∫

Ep(t, x, y)πR(y)m(dy) = e−λRtπR(x),

∫Ep(t, y, x)πR(y)m(dy) = e−λRtπR(x)

with some λR ≥ 0 for each t > 0 and every x ∈ E.(ii) Let C :=

∫E πR(x)πR(x)m(dx). There exist constants b, γ > 0 and T ′ > 0 such that for

t ≥ T ′ we have the estimate for the density

|CeλRtp(t, x, y)− πR(x)πR(y)| ≤ be−γt, x, y ∈ E.

(iii) The process X is recurrent under QπR defined by QπR |Ft= MπR

t Q∣∣∣Ft

with

MπRt = e−

∫ t0 r(Xs)ds+λRt πR(Xt)

πR(X0)

for each x ∈ E. Moreover, X is positive recurrent under QπR with the stationary distributionµ(dx) = C−1πR(x)πR(x)m(dx).(iv) If in addition m is a finite measure, m(E) < ∞, then the zero-coupon bond has the followinglong maturity estimate:

P (x, t) =

∫Ep(t, x, y)m(dy) = c0πR(x)e

−λRt +O(e−(γ+λR)t)

with c0 =∫E(1/πR(x))µ(dx) = C−1

∫E πR(x)m(dx) for each x ∈ E.

20

The proof is given in Qin and Linetsky (2014) and is based on Zhang et al. (2013) which,in turn, is based on Jentzsch’s theorem, a counterpart of Perron-Frobenius theorem for integraloperators in L2 spaces. (A further extension of Jentzsch’s theorem to operators in abstractBanach spaces is provided by the Krein-Rutman theorem).

In the special case when Pt = Pt, i.e. the pricing operators are symmetric with respect tothe measure m, (Pt)t≥0 can be interpreted as the transition semigroup of a symmetric Markovprocess Xr killed at the rate r (cf. Chen and Fukushima (2011) and Fukushima et al. (2010)).In particular, essentially all one-dimensional diffusions are symmetric Markov processes with thespeed measure m acting as the symmetry measure. In the symmetric case, Assumption 8.2 (ii)implies that for each t > 0 the pricing operator Pt is a symmetric Hilbert-Schmidt operatorin L2(E,m). It further implies that the pricing semigroup is trace class (cf. Davies (2007)Section 7.2) and, hence, for each t > 0 the pricing operator Pt has a purely discrete spectrume−λnt, n = 1, 2, . . . with 0 ≤ λ1 ≤ λ2 ≤ . . . repeated according to the eigenvalue multiplicitywith the finite trace

trPt =

∫Ep(t, x, x)m(dx) =

∞∑n=1

e−λnt < ∞.

Using the symmetry of the density, p(t, x, y) = p(t, y, x), and the Chapman-Kolmogorov equa-tion, Assumption 8.2 (iii) reduces to the assumption that there exists a constant T > 0 suchthat

supx∈E

p(t, x, x) < ∞, ∀t ≥ T.

Under the recurrent eigen-measure QπR ensured to exist under assumptions of Theorem 8.2,the transition function of X is given by

QπR(t, x, dy) = eλRt πR(y)

πR(x)p(t, x, y)m(dy)

and possesses the stationary measure

ς(dy) = C−1πR(y)πR(y)m(dy).

Theorem 8.3. In addition to assumptions in Theorem 8.2 assume that m is a finite measure,i.e. m(E) < ∞. Then the long-term measure exists and is identified with the recurrent eigen-measure, i.e. L = QπR .

The proof is by verifying Assumption 8.1 and, hence, establishing that in this case Theorem8.1 applies. For any function f such that |f | ≤ 1 and t ≥ t0 we have∣∣∣∣EQπR

x

[f(Xt)

πR(Xt)

]−

∫E

f(y)

πR(y)ς(dy)

∣∣∣∣ =

∣∣∣∣∫E(QπR(t, x, dy)− ς(dy))

f(y)

πR(y)

∣∣∣∣≤

∫E|QπR(t, x, dy)− ς(dy)| 1

πR(y)

=

∫EC−1 πR(y)

πR(x)m(dy)|CeλRtp(t, x, y)− πR(x)πR(y)|

1

πR(y)

≤ 1

πR(x)

∫EC−1m(dy)be−γt.

This verifies (8.2) with α = γ and c = C−1b∫E m(dy). We stress that assuming m is finite also

ensure P Tt = PT−t1(Xt) < ∞ for all T > t ≥ 0 and thus consistent with our general Assumption

2.1.

21

8.2.2 Cox-Ingersoll-Ross Model

Consider a CIR diffusiondXt = (a− κXt)dt+ σ

√XtdW

Qt , (8.3)

where a > 0, σ > 0 and κ ∈ R. In the CIR model the short rate is taken to be rt = Xt. Weinterpret (8.3) as the risk-neutral dynamics. The closed form expression for the density p(t, x, y)of the pricing kernel Pt is available (cf. Cox et al. (1985)). Qin and Linetsky (2014) (Section6.1.1) verify that it satisfies all assumptions in Theorem 8.3 and thus the long forward measure Lexists in the CIR model and coincides with the recurrent eigen-measure QπR . Qin and Linetsky(2014) determine the recurrent eigenfunction

πR(x) = eκ−γ

σ2 x

and the corresponding eigenvalue

λR =a

σ2(γ − κ),

where γ =√κ2 + 2σ2. From their results, the long bond in the CIR model is:

B∞t = eλRt πR(Xt)

πR(X0)= e

aσ2 (γ−κ)t+κ−γ

σ2 (Xt−X0).

(This expression can also be obtained directly by taking the limit T → ∞ of the ratio P (Xt, T −t)/P (X0, T ), where P (x, τ) is the CIR bond pricing function.) The L-measure dynamics of X

can be obtained from Girsanov’s theorem by using the Q-martingale B∞t /At = e−

∫ t0 XsdsB∞

t tochange measure from Q to L:

dXt = (a− γXt)dt+ σ√

XtdWLt .

We observe that the rate of mean reversion under L is higher than under Q: γ =√κ2 + 2σ2 > κ.

8.2.3 Vasicek Model

Consider an Ornstein-Uhlenbeck (OU) process

dXt = κ(θ −Xt)dt+ σdWQt (8.4)

with κ > 0, σ > 0. In the Vasicek model the short rate is taken to be rt = Xt. We interpret (8.4)as the risk-neutral dynamics. Qin and Linetsky (2014) determine the recurrent eigenfunction

πR(x) = e−xκ and the corresponding eigenvalue λR = θ − σ2

2κ2 in the Vasicek model, and obtainthe QπR-dynamics:

dXt = κ(θ − σ2

κ2−Xt)dt+ σdWQπR

t . (8.5)

In contrast to CIR, in the Vasicek model the short rate is not non-negative, and we cannotapply Theorem 8.3. Nevertheless, in Appendix E.1 we directly verify that the L1 convergencecondition in Theorem 4.1 holds in Vasicek model and, hence, the long bond exists and is givenby

B∞t = eλRt πR(Xt)

πR(X0)= e(θ−

σ2

2κ2)t− 1

κ(Xt−X0),

and, thus, the long forward measure L can be identified with QπR in the Vasicek model. Thus,we identify the L-dynamics with (8.5). We note that the long-run mean θ − σ2

κ2 < θ is lowerunder L than under Q in the Vasicek model.

22

8.2.4 A Square-root Model with Absorption at Zero: L Exists, QπR Does Not Exist

Consider a degenerate CIR model

dXt = −κXtdt+ σ√XtdW

Qt , (8.6)

where σ > 0 and κ ∈ R. The parameter a in the CIR SDE of Section 8.2.2 now vanishes,and the process has an absorbing boundary at zero. The short rate is taken to be rt = Xt. Weinterpret (8.6) as the risk-neutral dynamics. It is clear that under any locally equivalent measurethe boundary at zero remains absorbing and, thus, no recurrent eigenfunction exists. However,Appendix E.2 shows that Theorem 4.1 nevertheless applies, and the long bond is given by:

B∞t = e

κ−γ

σ2 (Xt−X0),

where γ =√κ2 + 2σ2. The L-measure dynamics of X can then be immediately obtained from

Girsanov’s theorem by using the Q-martingale B∞t /At = e−

∫ t0 XsdsB∞

t to change from Q to L:

dXt = −γXtdt+ σ√

XtdWLt

with γ =√κ2 + 2σ2. This example is interesting as it illustrates existence of the long bond,

long forward measure and, hence, the long-term factorization in a transient case. In this casethe recurrent eigen-measure QπR does not exist, while the long forward measure L neverthelessexists, but X is transient under L.

9 A Semimartingale Perspective on Ross’ Recovery

We conclude with an application to the Recovery Theorem of Ross (2013). Ross asks an inter-esting question: under what assumptions can one uniquely recover the market’s beliefs aboutphysical probabilities from Arrow-Debreu state prices implied by observed market prices ofderivative securities, such as options? Ross’ Recovery Theorem provides the following answer.If all uncertainty in an arbitrage-free, frictionless economy follows a finite state, discrete timeirreducible Markov chain and the pricing kernel satisfies a structural assumption of transitionindependence, then there exists a unique recovery of physical probabilities from given Arrow-Debreu state prices. Ross’ proof crucially relies on the Perron-Frobenius theorem establishingexistence and uniqueness of a positive eigenvector of an irreducible non-negative matrix. Carrand Yu (2012) observe that Ross’ recovery result can be extended to 1D diffusions on boundedintervals with regular boundaries at both ends by observing that the infinitesimal generator ofsuch a diffusion is a regular Sturm-Liouville operator that has a unique positive eigenfuction.They then rely on the regular Sturm-Liouville theory to show uniqueness of Ross recovery inthat setting.

Hansen and Scheinkman (2014) and Borovicka et al. (2014) point out that Ross’ assumptionof transition independence of the Markovian pricing kernel is a specialization of the factorizationin Hansen and Scheinkman (2009) to the case where the martingale component is degenerate,Mπ = 1. Qin and Linetsky (2014) investigate general continuous-time Markovian environmentswith all uncertainty driven by a Borel right process and also show that transition independencerestricts the pricing kernel to the form (8.1) for some positive eigenfunction π(x) with thecorresponding eigenvalue λ and with the degenerate martingale Mπ = 1. Borovicka et al.

23

(2014) further point out that under ergodicity assumptions that both fix uniqueness of a positiveeigenfunction and identify the corresponding eigen-measure Qπ with the measure we call in thepresent paper the long-term forward measure L, the transition independence assumption leadsto the identification of the physical measure P with the long forward measure L. Martin andRoss (2013), working in discrete time, ergodic finite-state Markov chain environments, presentsimilar results.

This previous literature crucially relies on the Markov property. We now discuss Ross’Recovery Theorem from the semimartingale perspective of this paper that does not require theMarkov assumption. Since the economic implication of transition independence of the pricingkernel in Ross’ discrete-time, finite state irreducible Markov chain economy is the identificationMπ = 1 and, consequently, the identification P = L, from the perspective of this paper Ross’assumption can be directly extended to semimartingale models by restricting the pricing kernelto what we called long-term risk-neutral in Definition 4.3 of Section 4. In other words, Ross’pricing kernel discounts at the rate of return on the long bond, assumes away the martingaleM∞

t = 1, and, hence, identifies P = L. By Proposition 4.1, an immediate economic consequenceof this restriction on the pricing kernel is the growth optimality of the long bond. In particular,we can immediately extend Ross’ recovery to HJM models by setting the martingale (6.9) tounity and, thus, identifying the market price of risk in the HJM model with the volatility of thelong bond:

γt = σ∞t .

In other words, if we start with a given risk-neutral HJM model under Q, as long as the HJMvolatility structure satisfies sufficient conditions in Section 6, Theorem 6.2 explicitly identifiesthe long forward measure L. Then Ross’ recovery further identifies P = L. We thus extendRoss’ recovery and Borovicka et al. (2014) interpretation of it to semimartingale environmentswithout assuming the Markov property.

We point out that Alvarez-Jermann and Bakshi-Chabi-Yo bounds on the components ofthe long-term factorization (extended to semimartingale enviroments in Theorems 5.2 and 5.3)allow direct empirical testing of the magnitude of the volatility of the martingale componentM∞ in the long term factorization. In particular, they give estimates of the magnitude of thespecification error resulting from assuming away the martingaleM∞

t = 1. Borovicka et al. (2014)point out that the empirical results of Alvarez and Jermann (2005) and Bakshi and Chabi-Yo(2012) imply that the martingale component is highly non-trivial, and neglecting it would leadto substantial model misspecification. Further empirical investigations of specification errorsresulting from the identification P = L could address the issues of selecting an appropriateempirically observable proxy for the long bond (thirty year maturity of US Treasuries may notbe long enough to provide a sufficiently good proxy for the long bond) that arises in testing Ross’recovery already in the setting of bond market returns and, further, a possible “survivorshipbias” of the high U.S. equity premiums observed in the 1900s to the 2000s sample.

10 Conclusion

This paper extends the long-term factorization of the pricing kernel due to Alvarez and Jermann(2005) in discrete-time ergodic environments and Hansen and Scheinkman (2009) in continu-ous ergodic Markovian environments to general semimartingale environments without assumingMarkov property. We show how the long bond and the long forward measure can be defined

24

in semimartingale environments, extend Alvarez and Jermann (2005) and Bakshi and Chabi-Yo(2012) bounds, obtain explicit long-term factorizations in (generally non-Markovian) Heath-Jarrow-Morton models and models of social discounting, provide sufficient conditions for exis-tence of the long-term factorization in Markovian models, complementing the results of Hansenand Scheinkman (2009), and extend to general semimartingale environments the interpretationof Ross (2013) recovery by Borovicka et al. (2014).

A Proof of Theorem 4.1

We first recall some results about semimartingale convergence originally introduced by Emery(1979) (see Czichowsky and Schweizer (2011), Kardaras (2013) and Cuchiero and Teichmann(2014) for recent applications of Emery’s semimartingale topology in mathematical finance).Emery’s distance between two semimartingales X and Y is defined by

dS(X,Y ) =∑n≥1

2−n sup|η|≤1

(E[1 ∧ |η0(X0 − Y0) +

∫ n

0ηsd(X − Y )s|]

), (A.1)

where the supremum is taken over all predictable processes η bounded by one, |ηt| ≤ 1 and∫ t0 ηsdXs denotes the stochastic integral of a predictable process η with respect to a semimartin-gale X (cf. Jacod and Shiryaev (2003)). When we wish to emphasize the probability measure,we write dPS(X,Y ). With this metric, the space of semimartingales is a complete topologicalvector space, and the corresponding topology is called the semimartingale topology. We denote

convergence in the semimartingale topology byS−→. By Theorem II.5 in Memin (1980), the

semimartingale convergence is invariant under locally equivalent measure transformations.The semimartingale topology is stronger than the topology of uniform convergence in prob-

ability on compacts (ucp), for which η only takes the form η(x) = 1[0,s](x) in (A.1):

ducp(X,Y ) =∑n≥1

2−nE[1 ∧ sups≤n

|Xs − Ys|].

The following inequality due to Burkholder is useful for proving convergence in the semi-martingale topology in Theorem 4.1 (see Meyer (1972) Theorem 47, p.50 for discrete martingalesand Cuchiero and Teichmann (2014) for continuous martingales, where a proof is provided insidethe proof of their Lemma 4.7).

Lemma A.1. For every martingale X and every predictable process η bounded by one, |ηt| ≤ 1,it holds that

aP( sups∈[0,t]

|∫ s

0ηudXu| > a) ≤ 18EP[|Xt|]

for all a ≥ 0 and t > 0.

We will also use the following result (cf. Kardaras (2013), Proposition 2.10).

Lemma A.2. If Xn S−→ X and Y n S−→ Y , then XnY n S−→ XY .

We will also make use of the following lemma.

Lemma A.3. If (Xnt )t≥0 is a family of martingales and there exists a process (X∞

t )t≥0 such

that for each t Xnt

L1

−→ X∞t as n → ∞, then (X∞

t )t≥0 is a martingale.

25

Proof. By Jensen’s inequality, for t > s,∣∣∣E[Xnt −X∞

t |Fs]∣∣∣ ≤ E[|Xn

t −X∞t ||Fs].

Thus, for each t and s such that t > s

E[|Xns − E[X∞

t |Fs]|] = E[∣∣∣E[Xn

t −X∞t |Fs]

∣∣∣] ≤ E [E[|Xnt −X∞

t ||Fs]] = E[|Xnt −X∞

t |].

Since Xnt

L1

−→ X∞t as n → ∞, above inequality implies that Xn

sL1

−→ E[X∞t |Fs] as n → ∞. On

the other hand, it also holds that Xns

L1

−→ X∞s as n → ∞. Thus X∞

s = E[X∞t |Fs]. This verifies

(X∞t )t≥0 is a martingale. 2

We are now ready to prove Theorem 4.1.(i) Since (MT

t )t≥0 are positive P-martingales with MT0 = 1, and for each t ≥ 0 random variables

MTt converge to M∞

t > 0 in L1, by Lemma A.3 (M∞t )t≥0 is also a positive P-martingale with

M∞0 = 1. Emery’s distance between the martingales MT for some T > 0 and M∞ is

dPS(MT ,M∞) =

∑n≥1

2−n sup|η|≤1

(EP[1 ∧ |

∫ n

0ηsd(M

T −M∞)s|]).

To prove MT S−→ M∞, it is suffice to prove that for all n

limT→∞

sup|η|≤1

(EP[1 ∧ |

∫ n

0ηsd(M

T −M∞)s|])= 0. (A.2)

We can write for an arbitrary ϵ > 0 (for any random variable X it holds that E[1 ∧ |X|] ≤P(|X| > ϵ) + ϵ)

EP[1 ∧ |∫ n

0ηsd(M

T −M∞)s|] ≤ P(|∫ n

0ηsd(M

T −M∞)s| > ϵ) + ϵ.

By Lemma A.1,

sup|η|≤1

EP[1 ∧ |∫ n

0ηsd(M

T −M∞)s|] ≤18

ϵEP[|MT

n −M∞n |] + ϵ.

Since limT→∞ EP[|MTn − M∞

n |] = 0, and ϵ can be taken arbitrarily small, Eq.(A.2) is verified

and, hence, MT S−→ M∞ (under P, as well as under any locally equivalent measure).

(ii) We have shown that MTt = StB

∞t

S−→ M∞t = StB

∞t . By Lemma A.2, BT

tS−→ B∞

t , and B∞t is

the long bond according to Definition 4.1 (the semimartingale convergence is stronger than theucp convergence).Part (iii) is a direct consequence of (i) and (ii).(iv) Define a new probability measure Q∞ by Q∞|Ft = M∞

t P|Ft for each t ≥ 0. The distancein total variation between the measures QT for some T > 0 and Q∞ on Ft is:

2 supA∈Ft

|QT (A)−Q∞(A)|.

26

For each t ≥ 0 we can write:

0 = limT→∞

EP[|MTt −M∞

t |] = limT→∞

EQ∞[∣∣∣BT

t

B∞t

− 1∣∣∣] .

Thus,

limT→∞

supA∈Ft

∣∣∣EQ∞[BT

t

B∞t

1A

]− EQ∞

[1A]∣∣∣ = 0.

SincedQT

dQ∞

∣∣∣∣Ft

=BT

t

B∞t

,

it follows thatlimT→∞

supA∈Ft

∣∣∣EQT[1A]− EQ∞

[1A]∣∣∣ = 0.

Thus QT converge to Q∞ in total variation on Ft for each t. Since convergence in total vari-ation implies strong convergence of measures, this shows that Q∞ is the long forward measureaccording to our Definition 4.2, Q∞ = L. 2

B Proofs of Pricing Kernel Bounds

Proof of Theorem 5.1. Let V denote the wealth process of a self-financing portfolio such thatStVt is a martingale. By Cauchy-Schwarz inequality we can write:

σ(ST

St

)σ(VT

Vt

)≥

∣∣∣∣E [(ST

St− E

[ST

St

])(VT

Vt− E

[VT

Vt

])]∣∣∣∣ . (B.1)

Since E[STVTStVt

]= 1 by the martingale property, we have

E[(

ST

St− E

[ST

St

])(VT

Vt− E

[VT

Vt

])]= 1− E[P T

t ]E[VT

Vt

].

(i) then immediately follows by substituting this result into (B.1), dividing both sides by

σ(VTVt

)E[STSt

]and recognizing that E

[STSt

]= E[P T

t ]. The conditional versions (ii) can be derived

in the same way. 2

Proof of Theorem 5.2. The proof is similar to the proof of Hansen-Jagannathan bounds. ByCauchy-Schwarz inequality,

σ(M∞

T

M∞t

)σ( VT /Vt

B∞T /B∞

t

)≥

∣∣∣E[(M∞T

M∞t

− E[M∞

T

M∞t

])( VT /Vt

B∞T /B∞

t

− E[ VT /Vt

B∞T /B∞

t

])]∣∣∣.Since E[STVT

StVt] = E[M

∞T VTB∞

t

M∞t VtB∞

T] = 1 and E[M∞

T /M∞t ] = E[Et[M

∞T /M∞

t ]] = 1, we have

E[(M∞

T

M∞t

−E[M∞

T

M∞t

])( VT /Vt

B∞T /B∞

t

−E[ VT /Vt

B∞T /B∞

t

])]= 1−E

[M∞T

M∞t

]E[ VT /Vt

B∞T /B∞

t

]= 1−E

[ VT /Vt

B∞T /B∞

t

],

27

and (i) immediately follows.

(ii)-(vi) can be proved in the same way. Note that by Cauchy-Shwarsz inequality E[B∞

tB∞

T

]E[B∞

TB∞

t

]≥

1. This allows us to drop the absolute value in (ii) and similarly in (v).2

Proof of Theorem 5.3. By the martingale property of StVt for the wealth process Vt of anyself-financing portfolio and Jensen’s inequality we have:

0 = logEt[STVT

StVt] ≥ Et[log

STVT

StVt] = Et[log

M∞T

M∞t

] + Et[logVTB

∞t

VtB∞T

]. (B.2)

Since M∞ is a martingale, Lt(M∞T /M∞

t ) = −Et[log(M∞T /M∞

t )] and from (B.2) we immediatelyobtain (ii). (i) is the unconditional version of (ii).

To prove (iii), we express L(STSt

) in terms of L(M∞

TM∞

t) and then use (i). By definition,

Lt(ST

St) = logEt[

ST

St]− Et[log

M∞T B∞

t

B∞T M∞

t

]

= logP Tt − Et[log

B∞t

B∞T

] + Lt(M∞

T

M∞t

)

= Et[logP Tt B∞

T

B∞t

] + Lt(M∞

T

M∞t

).

Taking the expectation on both sides yields

ELt(ST

St) = E[log

P Tt B∞

T

B∞t

] + ELt(M∞

T

M∞t

).

Combining it with the fact that L(X) = ELt(X) + L(Et[X]) (twice), we have

L(ST

St) = ELt(

ST

St) + L(Et[

ST

St])

= E[logP Tt B∞

T

B∞t

] + ELt(M∞

T

M∞t

) + L(P Tt )

= E[logP Tt B∞

T

B∞t

] + L(M∞

T

M∞t

)− L(Et[M∞

T

M∞t

]) + L(P Tt )

= E[logP Tt B∞

T

B∞t

] + L(M∞

T

M∞t

) + L(P Tt ).

Thus,

L(M∞

TM∞

t)

L(STSt

)=

L(M∞

TM∞

t)

L(M∞

TM∞

t) + L(P T

t ) + E[log PTt B∞

TB∞

t].

Using (i) and the assumption that E[log PTt VT

Vt] + L(P T

t ) > 0, if L(P Tt ) + E[log PT

t B∞T

B∞t

] ≥ 0,

1 ≥L(

M∞T

M∞t)

L(STSt

)≥

E[log

VTB∞t

VtB∞T

]E[log

PTt VT

Vt

]+ L(P T

t ).

28

If L(P Tt ) + E[log PT

t B∞T

B∞t

] < 0,

1 ≤L(

M∞T

M∞t)

L(STSt

)≤

E[log

VTB∞t

VtB∞T

]E[log

PTt VT

Vt

]+ L(P T

t ).

Combining these two cases yields (iii). (iv) can be shown similarly. 2

C Proofs for Section 6

Proof of Theorem 6.1. (i) We first consider the risk-neutral case with γ = 0:

dft = (Aft + αHJM(t, ω, ft))dt+∑j∈N

σj(t, ω, ft)dWQ,jt . (C.1)

By Assumption 6.1, αHJM(t, ω, h) is Lipschitz continuous in h and uniformly bounded (cf. Fil-ipovic (2001) Lemma 5.2.2). Thus by Theorem 2.4.1 of Filipovic (2001), Eq.(C.1) has a uniquecontinuous weak solution.

Uniqueness of a weak solution with non-zero γ follows by the application of Girsanov’stheorem. We already have uniqueness of a weak solution with γ = 0. By (6.4), γ satisfies theNovikov’s condition (cf. Filipovic (2001) Lemma 2.3.2). Thus, we can define a new measure Pby

P|Ft = exp(− 1

2

∫ t

0∥γs∥2ℓ2ds−

∫ t

0γs · dWQ

s

)Q|Ft .

Then by Girsanov’s theorem for infinite-dimensional Brownian motion (cf. Filipovic (2001))

W Pt = WQ

t +

∫ t

0γsds (C.2)

is an infinite-dimensional standard Brownian motions under P. Thus, ft is a unique weak solutionof the HJM equation (6.2) under P with general γ.

(ii) By Filipovic (2001) Theorem 5.2.1, zero-coupon bond price processes (P Tt /At)t≥0 taken

relative to the process At = e∫ t0 fs(0)ds are Q-martingales. This immediately yields the risk-

neutral factorization of the pricing kernel under P.(iii) Filipovic (2001) Eq.(4.17) gives

P Tt

P T0

= At exp(−

∫ t

0σTs · dWQ

s − 1

2

∫ t

0∥σT

s ∥2ℓ2ds).

Using Eq.(C.2) gives the bond dynamics under P. 2

Proof of Theorem 6.2. The proof consists of two parts. We first prove that the process on theright hand side of (6.8) is well defined (the integrals in the exponential are well defined). Nextwe prove that Eq.(4.1) in Theorem 4.1 holds with M∞

t = StB∞t and B∞

t defined by the righthand side of (6.8). This ensures that the process defined by the right hand side of (6.8) is thelong bond B∞

t and L defined by dLdP |Ft = M∞

t is the long forward measure. The expression forWL

t then follows from Girsanov’s Theorem (cf. Filipovic (2001) Theorem 2.3.3), and the SDEfor ft under L then follows immediately.

We first prove the following lemma which is central to all of the subsequent estimates.

29

Lemma C.1. The following estimate holds for any function h ∈ H0w′:∫ ∞

T|h(x)|dx ≤ C(T )∥h∥w′ , where C(T ) = K(T−ϵ/2 ∧ 1)

for some K > 0 and ϵ > 0.

Proof. Since w′(x) ≥ 1, for all h ∈ H0w′ we can write

|h(x)| = |∫ ∞

xh′(s)ds| ≤ ∥h∥w′(

∫ ∞

x

ds

w′(s))1/2 ≤ ∥h∥w′

( ∫ ∞

xK(s−(3+ϵ)∧1)ds

)1/2 ≤ ∥h∥w′K(x−(1+ϵ/2)∧1),

where the constant K can change from step to step. Thus∫∞T |h(x)|dx < ∥h∥w′K(T−ϵ/2∧1).

The next lemma ensures that the RHS of (6.8) is well defined.

Lemma C.2.∫ t0 ∥σ

∞s ∥2ℓ2ds ≤ C2(0)tD2

2.

Proof. By Lemma C.1,∫∞0 |σj

s(u)|du ≤ C(0)∥σjs∥Hw . This implies∫ t

0∥σ∞

s ∥2ℓ2ds ≤∫ t

0

∑j∈N

[ ∫ ∞

0|σj

s(u)|du]2ds

≤∫ t

0

∑j∈N

C2(0)∥σjs∥2w′ds

= C2(0)

∫ t

0∥σs∥2L0

2(Hw′ )ds

≤ C2(0)tD22.

By above lemma, the last integral in (6.8) is well defined. The stochastic integral∫ t0 σ

∞s ·dW P

s

is well defined due to Ito’s isometry. The first integral is bounded by

1

2

∫ t

0

(∥γs∥2ℓ2 + ∥σ∞

s ∥2ℓ2)ds ≤ 1

2

∫ t

0Γ(s)2ds+

1

2C2(0)tD2

2,

which is well defined by the fact that Γ ∈ L2(R+). Thus the right hand side of (6.8) is welldefined.

We now re-writePTt

PT0

and B∞t in terms of Q-Brownian motion WQ

t :

P Tt

P T0

= At exp(−

∫ t

0σTs · dWQ

s − 1

2

∫ t

0∥σT

s ∥2ℓ2ds),

B∞t = At exp

(−

∫ t

0σ∞s · dWQ

s − 1

2

∫ t

0∥σ∞

s ∥2ℓ2ds).

Fix the current t ≥ 0. By Remark 4.1, showing Eq.(4.1) is equivalent to showing that

limT→∞

EQ[∣∣∣∣ P T

t

P T0 At

− B∞t

At

∣∣∣∣] = 0. (C.3)

30

We first introduce some notation. For v ∈ [0, t] and T ∈ [t,∞] define

jTv :=

∫ v

0σTs · dWQ

s , kTv :=1

2

∫ v

0∥σT

s ∥2ℓ2ds,

σTv :=

∫ ∞

T−vσv(u)du, zTv :=

1

2

∫ v

0∥σT

s ∥2ℓ2ds, Y Tv := e−(jTv −j∞v )−zTv .

For p ≥ 1 and a random variable X we denote

∥X∥p :=(EQ[|X|p]

)1/p,

as long as the expectation is well defined. Then Eq.(C.3) can be re-written as

limT→∞

∥e−jTt −kTt − e−j∞t −k∞t ∥1 = 0.

By Holder’s inequality,

limT→∞

∥e−jTt −kTt − e−j∞t −k∞t ∥1 ≤ limT→∞

∥e−j∞t −k∞t ∥2∥e−(jTt −j∞t )−(kTt −k∞t ) − 1∥2.

Lemma C.3 and C.4 below show that ∥e−j∞t −k∞t ∥2 is finite and limT→∞ ∥e−(jTt −j∞t )−(kTt −k∞t ) −1∥2 = 0, respectively.

Lemma C.3. For each t > 0, there exists C such that

supv≤t

∥Y Tv ∥2 ≤ C < ∞ and ∥e−j∞t −k∞t ∥2 ≤ C < ∞.

Proof. We begin by considering the process (Y Tv )2 = e−(2jTv −2j∞v )−4zTv +2zTv for t ∈ [0, T ]. By Ito’s

formula, e−(2jTv −2j∞v )−4zTv is a local martingale. Since it is also positive, it is a supermartingale(in fact, it is a true martingale due to Lemma C.2 and Novikov’s criterion). Therefore for allv ≤ t,

EQ[e−(2jTv −2j∞v )−4zTv ] ≤ 1.

By Lemma C.2, |zTv | ≤ |k∞v | ≤ |k∞t | ≤ 12C

2(0)tD22. Thus ∥Y T

v ∥22 = EQ[e−(2jTv −2j∞v )−4zTv +2zTv ] ≤eC

2(0)tD22 . This implies

supv≤t

∥Y Tv ∥2 ≤ e

12C2(0)tD2

2 .

Similarly, (e−j∞t −k∞t )2 = e−2j∞t −4k∞t +2k∞t . The process e−2j∞t −4k∞t is a supermartingale, and

k∞t ≤ 12C

2(0)tD22 (Lemma C.2). Thus, ∥e−j∞t −k∞t ∥2 ≤ e

12C2(0)tD2

2 .

Lemma C.4.limT→∞

∥e−(jTt −j∞t )−(kTt −k∞t ) − 1∥2 = 0. (C.4)

Proof. We need the following two intermediate lemmas.

Lemma C.5. For T ≥ t, supv≤t |kTv − k∞v | ≤ C(0)C(T − t)tD22.

31

Proof.

supv≤t

|kTv − k∞v | = supv≤t

∣∣∣∣∣∣12∑j∈N

∫ v

0

((∫ T−s

0+

∫ ∞

0

)σjs(u)du

)(∫ ∞

T−sσjs(u)du

)ds

∣∣∣∣∣∣≤

∑j∈N

∫ t

0

(∫ ∞

0|σj

s(u)|du)(∫ ∞

T−s|σj

s(u)|du)ds

≤∑j∈N

∫ t

0C(0)∥σj

s∥w′C(T − s)∥σjs∥w′ds

= C(0)C(T − t)

∫ t

0

∑j∈N

∥σjs∥2w′ds

= C(0)C(T − t)

∫ t

0∥σs∥2L0

2(Hw′ )ds

≤ C(0)C(T − t)tD22.

Lemma C.6.limT→∞

∥Y Tt − 1∥2 = 0. (C.5)

Proof. By Ito’s formula,

Y Tt = 1 +

∫ t

0Y Tv σT

v · dWQv .

By Ito’s isometry, we have

∥Y Tt − 1∥22 = EQ

(∫ t

0∥Y T

v σTv ∥2ℓ2dv

).

By Lemma C.1, |σT,jv | ≤ C(T − v)∥σj

v∥w′ . Thus

∥Y Tt − 1∥22 ≤ EQ

(∑j∈N

∫ t

0|Y T

v |2C2(T − v)∥σjv∥2w′dv

)≤ C2(T − t)EQ

(∫ t

0|Y T

v |2∑j∈N

∥σjv∥2w′dv

)= C2(T − t)EQ

(∫ t

0|Y T

v |2∥σv∥2L02(Hw′ )

dv)

≤ C2(T − t)EQ(∫ t

0|Y T

v |2D22dv

)≤ C2(T − t)D2

2

∫ t

0EQ(|Y T

v |2)dv

≤ C2(T − t)D22

∫ t

0C2dv (Lemma C.3)

= C2(T − t)D22C

2t.

Since limT→∞C(T − t) = 0, Eq.(C.5) is verified.

32

Now we return to the proof of Lemma C.4.

∥e−(jTt −j∞t )−(kTt −k∞t ) − 1∥2 = ∥Y Tt ez

Tt −(kTt −k∞t ) − 1∥2

≤ ∥(Y Tt − 1)ez

Tt −(kTt −k∞t )∥2 + ∥ezTt −(kTt −k∞t ) − 1∥2.

Recall that by Lemma C.5, |kTt − k∞t | ≤ C(0)C(T − t)tD22. Using the same approach as Lemma

C.2, we can show that |zTt | ≤ 12C

2(T − t)tD22. Thus, we have

∥e−(jTt −j∞t )−(kTt −k∞t ) − 1∥2 ≤ ∥Y Tt − 1∥2e

12C2(T−t)tD2

2+C(0)C(T−t)tD22

+e12C2(T−t)tD2

2+C(0)C(T−t)tD22 − 1

Finally Eq.(C.4) is verified using Lemma C.6 and the fact that limT→∞C(T − t) = 0.

D Proof of Theorem 8.1

Let QπR(t, x, ·) denote the transition measure of X under QπR . We first prove that Assumption8.1 implies

∫E ς(dy) 1

πR(y) < ∞. Since we assume EPx[St] < ∞, P t

0 < ∞ for all t > 0. Rewrite thezero-coupon bond price under QπR :

P t0 = e−λRtπR(x)EQπR

x [1

πR(Xt)] < ∞.

Thus,

EQπR

x [1

πR(Xt)] =

∫EQπR(t, x, dy)

1

πR(y)< ∞

for all t > 0 and x ∈ E. This implies∫E ς(dy) 1

πR(y) =

∫EQπR(t, x, dy)

1

πR(y)+

∫E

(ς(dy)−QπR(t, x, dy)

) 1

πR(y)dy

≤∫EQπR(t, x, dy)

1

πR(y)+

c

πR(x)e−αt < ∞.

We now verify Eq.(4.1) with M∞t = MπR

t . This then identifies eλRt πR(Xt)πR(X0)

with the long

bond B∞t and QπR = L. We choose to verify it under QπR , i.e. Eq.(4.3) with QV = QπR due to

its convenient form. Since P Tt = e−λR(T−t)πR(Xt)EQπR

Xt[ 1πR(XT−t)

], we have

e−λRtPTt πR(X0)

P T0 πR(Xt)

=EQπR

Xt[ 1πR(XT−t)

]

EQπR

X0[ 1πR(XT ) ]

.

Let J :=∫E ς(dy) 1

πR(y) . Since EQπR

x [ 1πR(Xt)

] =∫E QπR(t, x, dy) 1

πR(y) , by Eq.(8.2) we have:

J − c

πR(Xt)e−α(T−t) ≤ EQπR

Xt[

1

πR(XT−t)] ≤ J +

c

πR(Xt)e−α(T−t), (D.1)

and for each initial state X0 = x ∈ E:

J − c

πR(x)e−αT ≤ EQπR

x [1

πR(XT )≤ J +

c

πR(x)e−αT . (D.2)

33

For each x ∈ E there exists T0 such that for T ≥ T0,c

πR(x)e−αT ≤ J/2. We can thus write for

each x ∈ E:

−1 ≤ e−λRt P Tt πR(x)

P T0 πR(Xt)

− 1 ≤ 2

J

(c

πR(Xt)e−α(T−t) +

c

πR(x)e−αT

),

Thus, ∣∣∣∣e−λRt P Tt πR(x)

P T0 πR(Xt)

− 1

∣∣∣∣ ≤ 2

J

(c

πR(Xt)e−α(T−t) +

c

πR(x)e−αT

)+ 1.

Since for each t the Ft-measurable random variable 1πR(Xt)

is integrable under QπRx for each

x ∈ E, for each t the Ft-measurable random variable∣∣∣e−λRt PT

t πR(x)

PT0 πR(Xt)

− 1∣∣∣ is bounded by an

integrable random variable. Furthermore, by Eq.(D.1) and (D.2),

limT→∞

∣∣∣∣e−λRt P Tt (ω)πR(x)

P T0 πR(Xt(ω))

− 1

∣∣∣∣ = 0

for each ω. Thus, by the Dominated Convergence Theorem Eq.(4.3) is verified with B∞t =

eλRt πR(Xt)πR(X0)

. 2

E Verification of Theorem 4.1 for Markovian Examples

E.1 Vasicek Model

The bond price in Vasicek model is given by

P Tt = P (Xt, T − t) = A(T − t)e−XtB(T−t), (E.1)

where

B(τ) =1− e−κτ

κ, A(τ) = exp

(θ − σ2

2κ2)(B(τ)− τ)− σ2

4κB2(τ)

.

We can verify directly that

limT→∞

P (y, T − t)

P (x, T )= e(θ−

σ2

2κ2)t− 1

κ(y−x).

It remains to verify Eq.(4.3) under QπR with B∞t = e(θ−

σ2

2κ2)t− 1

κ(Xt−X0). By Eq.(E.1),

P Tt

P T0 B∞

t

= exp((θ

κ− σ2

2κ3)(e−κT − e−κ(T−t))− σ2

4κ3(e−2κ(T−t) − e−2κT − 2e−κ(T−t) + 2e−κT )

+Xte−κ(T−t)

κ−X0

e−κT

κ

).

It is easy to verify that

limT→∞

∣∣∣∣ P Tt (ω)

P T0 B∞

t (ω)− 1

∣∣∣∣ = 0 (E.2)

for each ω. On the other hand, for any constant ϵ > 0 there exists T0 > 0 such that for T ≥ T0∣∣∣∣ P Tt

P T0 B∞

t

− 1

∣∣∣∣ < (1 + ϵ)eϵXt + 1.

In Vasicek model Xt has a Gaussian distribution and thus eϵXt is integrable for all ϵ > 0. Thus,by Eq.(E.2) and the Dominated Convergence Theorem Eq.(4.3) holds under QV = QπR .

34

E.2 Square-root Model with Absorption at Zero

Consider a CIR process (8.5) under Q with a = 0. The CIR bond price is given by P Tt =

e−XtB(T−t) with B(t) = 2(etγ−1)2γ+(κ+γ)(etγ−1) and γ =

√κ2 + 2σ2. We can verify directly that

limT→∞

P (y, T − t)

P (x, T )= e

κ−γ

σ2 (y−x).

By the affine property of the CIR model, we can also show that π(x) = eκ−γ

σ2 x is an eigenfunctionof the CIR pricing semigroup with a = 0 with the eigenvalue λ = 0. We now verify Eq.(4.3)

under QV = Qπ with B∞t = e

κ−γ

σ2 (Xt−X0). First observe that

P Tt

P T0 B∞

t

= e

(γ−κ

σ2 −B(T−t))Xt−

(γ−κ

σ2 −B(T ))X0 .

Since

limτ→∞

(γ − κ

σ2−B(τ)

)= 0,

we have

limT→∞

∣∣∣∣ P Tt (ω)

P T0 Vt(ω)

− 1

∣∣∣∣ = 0 (E.3)

for each ω. On the other hand, for any constant ϵ > 0 there exists T0 > 0 such that for T ≥ T0∣∣∣∣ P Tt

P T0 Vt

− 1

∣∣∣∣ ≤ (1 + ϵ)eϵXt + 1.

By the affine property of the CIR model, there exists ϵ > 0 such that eϵXt is integrable. Thus,

by Eq.(E.3) and the Dominated Convergence Theorem Eq.(4.3) holds with B∞t = e

κ−γ

σ2 (Xt−X0)

and L = Qπ, even though π = eκ−γ

σ2 x is not a recurrent eigenfunction.

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