AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 438258 10.1155/2014/438258 438258 Research Article Integration by Parts and Martingale Representation for a Markov Chain http://orcid.org/0000-0003-2823-5138 Siu Tak Kuen 1,2 He Shuping 1 Cass Business School City University London 106 Bunhill Row, London EC1Y 8TZ UK city.ac.uk 2 Department of Applied Finance and Actuarial Studies Faculty of Business and Economics Macquarie University, Sydney, NSW 2109 Australia mq.edu.au 2014 262014 2014 30 10 2013 10 05 2014 2 6 2014 2014 Copyright © 2014 Tak Kuen Siu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Integration-by-parts formulas for functions of fundamental jump processes relating to a continuous-time, finite-state Markov chain are derived using Bismut's change of measures approach to Malliavin calculus. New expressions for the integrands in stochastic integrals corresponding to representations of martingales for the fundamental jump processes are derived using the integration-by-parts formulas. These results are then applied to hedge contingent claims in a Markov chain financial market, which provides a practical motivation for the developments of the integration-by-parts formulas and the martingale representations.

1. Introduction

Integration by parts is at the heart of Malliavin calculus and its applications. It is deemed to be useful in mathematical finance, stochastic filtering and control as well as the theory of partial differential equations. Particularly, in mathematical finance, an integration-by-parts formula is useful in hedging contingent claims, numerical computations of Greeks, and portfolio optimization; see, for example, Benth et al. , León et al. , Imkeller , and Fournié et al. [4, 5], amongst others. Indeed, integration-by-parts formulas are one of the key results in a number of works on Malliavin calculus for stochastic differential equations driven by Wiener processes and jump processes. Some examples are Bismut , Bichteler et al. , Bass and Cranston , Norris , and Elliott and Tsoi [10, 11] to name a few. These authors adopted the approach to Malliavin calculus pioneered by Bismut , where an integration-by-parts formula was established by first considering a “small’’ perturbation of the original process and then compensating the effect of the perturbation by Girsanov's change of measure. For an excellent account of Malliavin calculus and its applications, one may refer to, for example, Nualart , Privault , and di Nunno et al. .

Markov chain is an important mathematical tool in probability theory and has vast applications in diverse fields. For example, in finance and actuarial science, there has been an interest in pricing contingent claims under Markov chain markets; see, for example, Norberg  and Elliott and Kopp  for bond pricing in a Markov chain market, Song et al.  for pricing options in a multivariate Markov chain market, Elliott et al.  and van der Hoek and Elliott [19, 20] for pricing options in Markov chain markets, and Norberg  and Koller  for pricing insurance products in Markov chain models. In statistics, particularly in nonlinear time series analysis, Markov chain plays an important role in studying the stochastic stability and ergodicity of stochastic difference equations; see, for example, Tong . Markov chain also plays an important role in stochastic filtering and control. There is a large amount of literature on the use of Markov chain and related stochastic processes in stochastic filtering and control. Some recent literature is Shen et al. , He and Liu [25, 26], Zhang et al. , He , Siu , Ellliott and Siu , and Wu et al. , amongst others. The monograph by Elliott et al.  provided discussions on hidden Markov models and their applications in various fields such as signal processing and image processing. The monographs by Yin and Zhang [33, 34] provided discussions on the theories and applications of discrete-time and continuous-time Markov chain, respectively. A recent monograph by Ching et al.  presented applications of Markov chain in diverse fields such as manufacturing systems, marketing, and finance.

It appears that in the finance and actuarial science literature much attention has been given to pricing contingent claims in Markov chain markets. It seems that relatively less attention has been paid to hedging contingent claims in Markov chain markets. An integration-by-parts formula is a useful tool for hedging contingent claims. It seems that the literature mainly focuses on developing and applying integration-by-parts formulas in the cases of Wiener processes, Lévy processes, and single jump processes (see, e.g., Elliott and Tsoi [10, 11], Nualart , Privault , and di Nunno et al. ). An integration-by-parts formula in the case of a Markov chain seems lacking. Motivated by the hedging problem in Markov chain markets, it may be of interest to derive an integration-by-parts formula which is useful for hedging contingent claims in Markov chain markets.

In this paper, we derive integration-by-parts formulas for functions of a family of fundamental jump processes relating to a continuous-time, finite-state Markov chain using the Bismut measure change approach. The formulas are derived by considering “small’’ perturbations to the jump intensity parameters of the fundamental jump processes, which are then compensated by Girsanov's measure change. Using the integration-by-parts formulas, new expressions for the integrands in representations of martingales for the fundamental jump processes are derived. Firstly, we consider a function of the terminal values of the fundamental jump processes. Then, the results are extended to a function of the integrals with respect to the whole paths of the fundamental jump processes. The function of the path integrals may be considered a canonical form of a random variable which is measurable with respect to filtration generated by the whole path of the Markov chain. No infinite-dimensional calculus of variations is involved in the derivations. Indeed, only finite-dimensional calculus is adopted. The martingale representation results derived here may be useful for hedging contingent claims in the Markov chain financial market developed by Norberg , where the dynamics of share prices were driven by the basic martingales of the fundamental jump processes relating to a continuous-time, finite-state Markov chain.

The rest of the paper is organized as follows. Section 2 describes the Markov chain, the fundamental jump processes, and the basic martingales relating to the chain. Section 3 derives the integration-by-parts formula for a function of the terminal values of the fundamental jump processes. In Section 4, the expression of the integrand in the martingale representation is obtained. The results are then extended to a function of the integrals of the whole paths of the fundamental jump processes in Section 5. An application of the martingale representation result to hedging contingent claims in the Markov chain financial market of Norberg  is given in Section 6. Section 7 summarizes the paper and suggests some potential topics for future research.

2. Markov Chain, Fundamental Jump Processes and Basic Martingales

The aim of this section is to present some known results in Markov chain, its fundamental jump processes and basic martingales which are relevant to the later developments.

Consider a complete probability space (Ω,F,P) and a finite time horizon T[0,T], where T<. Let X{X(t)tT} be a continuous-time, finite-state Markov chain on (Ω,F,P). As in Elliott et al. , we suppose that the state space of the chain X is a finite set of standard unit vectors E{e1,e2,,eN} in RN, where the jth component of ei is the Kronecker delta δij, for each i,j=1,2,,N. The space E is called the canonical state space of X.

To specify the probability laws of the chain X, we define a family of rate matrices, or intensity matrices, {A(t)tT} under P, where, for each tT, A(t)[aij(t)]i,j=1,2,,N. For each i,j=1,2,,N with ij and each tT, aij(t) is the instantaneous transition intensity of the chain X from state ei to state ej at time t. Note that for each i,j=1,2,,N and each tT,

aij(t)0, for ij;

j=1Naij(t)=0, so aii(t)0.

We suppose here that, for each i,j=1,2,,N, aij(t) is a bounded and deterministic function of time t.

Let FX{FX(t)tT} be the P-augmentation of the natural filtration generated by the chain X. Note that FX is right-continuous. Then with the canonical state space of the chain X, Elliott et al.  obtained the following semimartingale dynamics for X: (1)X(t)=X(0)+0tA(u-)X(u)du+M(t),tT. Here M{M(t)tT} is an RN-valued, square-integrable, (FX,P)-martingale.

For each i,k=1,2,,N with ik, let Jik{Jik(t)tT}, where Jik(t) counts the number of transitions of the chain X from state ei to state ek up to and including time t. That is, (2)Jik(t)0<stX(s-),eiX(s),ek.{Jiki,k=1,2,,N,ik} is called a family of fundamental jump processes relating to the chain X; ·,· is the scalar product in RN.

Define, for each i,k=1,2,,N with ik, a process Mik{Mik(t)tT} by putting (3)Mik(t)0tX(s-),eidM(s),ek. Then it is obvious from the definition that Mik, i,k=1,2,,N, are (FX,P)-martingales and {Miki,k=1,2,,N,ik} is called a family of basic martingales. Indeed these martingales are orthogonal, purely discontinuous, and square-integrable. Furthermore, Mik(0)=0.

The following lemma gives the semimartingale decomposition for Jik. This result is standard (see, e.g., Elliott  and Elliott et al. ).

Lemma 1.

For each i,k=1,2,,N with ik and each tT, (4)Jik(t)=0taik(s)X(s),eids+Mik(t).

Proof.

The proof of this lemma is standard. For the sake of completeness, we present the proof here: (5)Jik(t)0<stX(s-),eiX(s),ek=0<stX(s-),eiΔX(s),ek=0tX(s-),eidX(s),ek=0tX(s-),eiA(s)X(s-),ekds+0tX(s-),eidM(s),ek=0taik(s)X(s-),eids+Mik(t)=0taik(s)X(s),eids+Mik(t). The last equality is due to the fact that the set of all jump times of the chain X has zero “dt’’-measure.

From Lemma 1 and the definition of Mik, (6)Mik(t)Jik(t)-0taik(s)X(s),eids,tT, is an (FX,P)-martingale. Consequently, under P, {aik(t)X(t),eitT} is the intensity process of Jik.

3. Integration by Parts for Functions of Fundamental Jump Processes

In this section we first present small perturbations to the jump intensities of the fundamental jump processes and then compensate the perturbations by a Girsanov-type measure change. The integration-by-parts formula for a “suitable’’ function of the terminal values of the fundamental jump processes is then derived by differentiation. The techniques used to derive the integration-by-parts formula here are adapted to those used in Elliott and Tsoi  for deriving an integration-by-parts formula for a single jump process. It seems that the origin of these techniques may be traced back to the work of Bismut .

For each i,k=1,2,,N with ik, let ηik{ηik(t)tT} be a nonnegative, P-a.s. bounded, FX-predictable process. Then for an arbitrarily small ϵ>0, we define a small “stochastic’’ perturbation aikϵ(t) to aik(t) in the direction ηik(t) by putting (7)aikϵ(t)(1+ϵηik(t))aik(t). We then take (8)aiiϵ(t)-k=1,ikNaikϵ(t), so that (9)k=1Naikϵ(t)=0. Note that, for each tT, ηik(t)>0 and ϵ>0, so aikϵ(t)0, ik, and aiiϵ(t)0.

Define, for each i,k=1,2,,N with ik, the jump process Jikϵ{Jikϵ(t)tT} by putting (10)Jikϵ(t)0taikϵ(t)X(u),eidu+Mik(t), where Mik(t) is defined in Section 2 as follows: (11)Mik(t)0tX(s-),eidM(s),ek. By definition, (12)Mik(t)=Jikϵ(t)-0taikϵ(t)X(u),eidu,tT, is an (FX,P)-martingale. Consequently, Jikϵ has the intensity process {aikϵ(t)X(t),eitT} under P and it is related to Jik as follows: (13)Jikϵ(t)Jik(t)+ϵ0tηik(u)aik(u)X(u),eidu. To simplify the notation, write λik(t)ηik(t)aik(t)X(t),ei, for each i,k=1,2,,N with ik and each tT. Then (14)Jikϵ(t)=Jik(t)+ϵ0tλik(u)du. The process Jikϵ is called a perturbed process of the fundamental jump process Jik, so we have a family of perturbed processes {Jikϵi,k=1,2,,N,ik} corresponding to the family of the fundamental jump processes {Jiki,k=1,2,,N,ik}.

For each i,k=1,2,,N with ik and each tT, let (15)θikϵ(t)-ϵηik(t)1+ϵηik(t).

Define, for each tT, (16)Zϵ(t)i,k=1,ikN0tθikϵ(u-)dMik(u). Consider an FX-adapted process Λϵ{Λϵ(t)tT} defined by setting (17)Λϵ(t)=1+0tΛϵ(u-)dZϵ(u). Then by Elliott  (see Theorem 13.5 therein), (18)Λϵ(t)=E(Zϵ)(t)=0<ut(1+ΔZϵ(u)), where E(Zϵ){E(Zϵ)(t)tT} is the stochastic exponential of the process Zϵ; ΔZϵ(t)Zϵ(t)-Zϵ(t-).

Then, for each tT, (19)Λϵ(t)=exp(i,k=1,ikN0t[ln(1+θikϵ(u))-θikϵ(u)]×aikϵ(u)X(u),eidu+i,k=1,ikN0tln(1+θikϵ(u))dMik(u)). Note that by definition θikϵ(t)>-1, for each tT, so the process Λϵ{Λϵ(t)tT} is strictly positive. Furthermore, Λϵ is an (FX,P)-martingale.

A new probability measure Pϵ equivalent to P on FX(T) is now defined by putting (20)dPϵdP|FX(T)Λϵ(T). The following lemma will be used to derive the integration-by-parts formula.

Lemma 2.

The Pϵ-law of Jikϵ, i,k=1,2,,N with ik, is equal to the P-law of Jik, i,k=1,2,,N with ik.

Proof.

By a version of Girsanov’s theorem, the process Mikϵ{Mikϵ(t)tT} defined by (21)Mikϵ(t)Jikϵ(t)-0t(1+θikϵ(u))aikϵ(u)X(u),eidu,tT, is an (FX,Pϵ)-martingale. Note that (22)(1+θikϵ(t))aikϵ(t)=aik(t), so Jikϵ has the intensity process {aik(t)X(t),eitT} under Pϵ. This is the same as the intensity process of Jik under P.

Remark 3.

The (FX,Pϵ)-martingale Mikϵ defined in the proof of Lemma 2 is related to the (FX,P)-martingale Mik as follows: (23)Mikϵ(t)=Mik(t)-ϵ0tλik(u)du,tT.

To simplify our notation and illustrate the main idea, we consider the situation where the chain X has two states. In this case, the family of fundamental jump processes relating to the chain is {J12,J21} and its corresponding perturbed processes are {J12ϵ,J21ϵ}.

Let G:R2R be any measurable, integrable, and differentiable function. Note that from Lemma 2 the Pϵ-law of (J12ϵ(T),J21ϵ(T)) is the same as the P-law of (J12(T),J21(T)). Consequently, (24)E[G(J12(T),J21(T))]=Eϵ[G(J12ϵ(T),J21ϵ(T))]. Here E and Eϵ are expectations under P and Pϵ, respectively.

Write (25)J(T)(J12(T),J21(T))R2,Jϵ(T)(J12ϵ(T),J21ϵ(T))R2, where y is the transpose of a vector, or a matrix, y.

Define the following gradient of G with respect to x(x1,x2)R2: (26)DxG(x)=(x1G(x),x2G(x))R2. Then the following theorem gives the integration-by-parts formula.

Theorem 4.

For each tT, let (27)M-(t)(M12(t),M21(t))R2,η(t)(η12(t),η21(t))R2. Write, for each tT, (28)φ  (t)(0tλ12(u)du,0tλ21(u)du)R2. Then for any measurable, integrable, and differentiable function G:R2R, (29)E[DxG(J(T)),φ(T)]=E[G(J(T))0Tη(u)dM-(u)].

Proof.

By a version of Bayes' rule, (30)E[G(J(T))]=Eϵ[G(Jϵ(T))]=E[Λϵ(T)G(Jϵ(T))]. Differentiating both sides with respect to ϵ and setting ϵ=0 give (31)E[ϵΛϵ(T)|ϵ=0G(Jϵ(T))|ϵ=0]+E[Λϵ(T)|ϵ=0DxG(Jϵ(T)),ϵJϵ(T)|ϵ=0]=0. It is obvious that Λϵ(T)|ϵ=0=1 and that Jϵ(T)|ϵ=0=J(T). Consequently, (32)E[ϵΛϵ(T)|ϵ=0G(J(T))]+E[DxG(J(T)),ϵJϵ(T)|ϵ=0]=0. Now (33)ϵJϵ(T)|ϵ=0=(0Tλ12(u)du,0Tλ21(u)du)=φ(T),ϵΛϵ(T)=Λϵ(T)×[i,k=1,ik20Tln(11+ϵηik(u))ηik(u)aik(u)I{X(u)=ei}dui,k=1,ik20Tln(11+ϵηik(u))ηik(u)aik(u)I{X(u)=ei}du-i,k=1,ik20Tηik(u)(1+ϵηik(u))2dMik(u)]. Then (34)ϵΛϵ(T)|ϵ=0=-i,k=1,ik20Tηik(u)dMik(u)=-0Tη(u)dM-(u). Hence the result follows.

The following two integration-by-parts formulas are immediate consequences of Theorem 4.

Corollary 5.

For any measurable, integrable, and differentiable function G:R2R, (35)E[x1G(J(T))0Tλ12(u)du]=E[G(J(T))0Tη12(u)dM12(u)].

Proof.

The result follows by putting η21(t)=0, for all tT, in Theorem 4.

Corollary 6.

For any measurable, integrable, and differentiable function G:R2R, (36)E[x2G(J(T))0Tλ21(u)du]=E[G(J(T))0Tη21(u)dM21(u)].

Proof.

The result follows by putting η12(t)=0, for all tT, in Theorem 4.

Remark 7.

The integration-by-parts formula in Corollary 5 (Corollary 6) may be interpreted as an integration-by-parts formula obtained by perturbing the intensity {a12(t)tT} ({a21(t)tT}) along the direction η12 (η21).

Remark 8.

In Elliott and Kohlmann , an integration-by-parts formula for functions of jump processes was developed. Using the concept of stochastic flows, the integration-by-parts formula was derived for functions of the terminal values of jump processes. An advantage of the approach by Elliott and Kohlmann  is that the integration-by-parts formula was derived without using infinite-dimensional calculus. The integration-by-parts formula for functions of the terminal values of jump processes has an important application. Elliott and Kohlmann  demonstrated how this integration-by-parts formula may be applied to establish the existence and smoothness of the density of a jump process. This is a key area of application of Malliavin calculus. Using the method in Elliott and Kohlmann , the integration-by-parts formula in Theorem 4 may be used to establish the existence and uniqueness of the densities of some stochastic processes depending on the fundamental jump processes relating to the chain. This may represent an interesting topic for future research.

Remark 9.

In the Markov chain financial market of Norberg , the dynamics of share prices are described by the fundamental jump processes relating to a continuous-time, finite-state Markov chain. The integration-by-parts formula in Theorem 4 may be used to hedge contingent claims whose payoffs depend on the terminal values of the share prices in the continuous-time Markov chain market of Norberg . We will discuss this in some detail in Section 6.

4. Martingale Representation Using Integration by Parts

Martingale representation is one of the fundamental results in stochastic analysis and calculus. It has many significant applications in diverse fields such as mathematical finance, stochastic filtering, and control. A crucial question in a martingale representation is to determine the integrand in the representation. This question is of primary importance in many applications of martingale representations. The Clark-Haussmann-Ocone-Karatzas formula was developed to address this question in the case of a Wiener space (see Clark , Haussmann , Ocone , Ocone and Karatzas , and Karatzas et al. ). Elliott and Kohlmann  pioneered the use of stochastic flows to identify the integrand in a stochastic integral in a martingale representation under a Markov diffusion setting. Elliott and Kohlmann  extended the approach in Elliott and Kohlmann  to the case of a Markov jump process. Elliott and Tsoi [10, 11] adopted integration-by-parts formulas to derive integrands in martingale representations in a single jump process and a Poisson process, respectively. Aase et al.  adopted a white-noise approach to Malliavin calculus to establish a white-noise generalization of the Clark-Haussmann-Ocone-Karatzas formula in the cases of multidimensional Gaussian white noise, multidimensional Poisson white noise, and their combination. Di Nunno et al.  adopted a chaos expansion approach to Malliavin calculus to establish a white-noise generalization of the Clark-Haussmann-Ocone-Karatzas formula for Lévy processes.

In this section, we apply the integration-by-parts formula obtained in the last section to derive the integrand in a martingale representation for a function of the terminal values of the fundamental jump processes. Though the techniques to be used here are similar to those adopted in Elliott and Tsoi [10, 11], it seems that the formulas of the integrand derived here appear to be new. Again to simplify our notation, we consider here the two-regime Markov chain presented in Section 3.

Note that the filtration FX generated by the chain X is the same as the filtration generated by the family of fundamental jump processes {J12,J21}. Then we state the following martingale representation result which was due to Brémaud .

Theorem 10.

For any real-valued, square-integrable (FX,P)-martingale L{L(t)tT}, (37)L(T)E[L(T)]+0Tγ(u)dM-(u), for some R2-valued, FX-predictable process {γ(t)tT}.

Furthermore, we need the following expression for the predictable quadratic variation {M,M(t)tT} of M{M(t)tT}, which was derived in Elliott et al. .

Lemma 11.

Let diag[y] be a diagonal matrix with the diagonal elements being given by the components in a vector y. For each tT, (38)M,M(t)=0t(diag[A(u)X(u)]-diag[X(u)]A(u)-A(u)diag[X(u)]diag[X(u)]A(u))du.

To simplify our notation, let {f(t)tT} be a matrix-valued process defined as follows: (39)f(t)diag[A(t)X(t)]-diag[X(t)]A(t)-A(t)diag[X(t)]R2R2. Note that {f(t)tT} is the density process of the measure dM,M(t) with respect to the Lebesgue measure dt on (T,B(T)) and dM,M(t) is absolutely continuous with respect to dt, where B(T) is the Borel σ-field generated by open subsets of T.

Then (40)M,M(t)=0tf(u)du. The following lemma will be used to derive the expressions for the integrand in the martingale representation.

Lemma 12.

For each i,k=1,2 with ik, the predictable quadratic variation of Mik, namely {Mik,Mik(t)tT}, is given by (41)Mik,Mik(t)=0tX(u),eiekf(u)ekduR.

Proof.

Recall that (42)Mik(t)0tX(u-),eidM(u),ek=0tX(u-),eiekdM(u). Then (43)Mik,Mik(t)=0tX(u-),eiekdM,M(u)ekX(u-),ei=0tX(u-),eiekf(u)ekdu=0tX(u),eiekf(u)ekdu. The last equality follows from the fact that the set of all jump times of the chain X has “dt’’-measure zero.

By the martingale representation presented in Theorem 10, (44)G(J(T))=E[G(J(T))]+0Tγ(u)dM-(u), for some FX-predictable process γ{γ(t)tT}.

It can be supposed that E[G(J(T))]=0 by subtraction. Then (45)G(J(T))=0Tγ(u)dM-(u).

The integrand γ is then determined in the following theorem. Though the techniques used in the proof of the following theorem are similar to those used in Proposition 3.5 of Elliott and Tsoi , the expressions for the integrand presented below appear to be new.

Theorem 13.

Suppose that a12(t),a21(t)>0 for each tT. Then the integrand γ{γ(t)tT}, where γ(t)(γ1(t),γ2(t))R2, is determined by (46)γ1(t)=E[x1G(J(T))FX(t-)]a12(t)a21(t),ontheset  {X(t)=e1},γ2(t)=E[x2G(J(T))FX(t-)]a21(t)a12(t),ontheset  {X(t)=e2}.

Proof.

We only give the proof for the integrand γ1(t) since the integrand γ2(t) can be derived similarly. Firstly, by the martingale representation for G(J(T)), Lemma 12, and the orthogonality of M12 and M21, (47)E[G(J(T))0Tη12(u)dM12(u)]=E[(0Tγ(u)dM-(u))(0Tη12(u)dM12(u))]=E[0Tγ1(u)η12(u)dM12,M12(u)]=E[0Tγ1(u)η12(u)e2f(u)e2I{X(u-)=e1}du]. Then using the integration-by-parts formula in Corollary 5, (48)E[0Tx1G(J(T))λ12(u)du]=E[0Tγ1(u)η12(u)e2f(u)e2I{X(u-)=e1}du]. For each uT, let (49)ψ(u)x1G(J(T))a12(u)I{X(u-)=e1}. Then there exists an FX-predictable projection {ψ*(u)uT} of {ψ(u)uT} such that, for each uT, (50)ψ*(u)=E[ψ(u)FX(u-)],P-a.s., so that (51)ψ*(u)=E[x1G(J(T))FX(u-)]a12(u)I{X(u-)=e1},P-a.s. Furthermore, for any FX-predictable process {K(u)uT}, (52)E[K(u)ψ(u)]=E[K(u)E[ψ(u)FX(u-)]]=E[K(u)ψ*(u)]. Write H for the family of subsets of T×Ω of the forms {0}×F0 and (u,t]×Fu, where F0FX(0) and FuFX(u) for 0u<tT. Note that the predictable σ-field on the product space T×Ω with respect to FX is generated by H.

We now take η12=I{0}×F0 or η12=I(u,t]×Fu, where I{0}×F0 and I(u,t]×Fu are the indicator functions of the events {0}×F0 and (u,t]×Fu, respectively. Then the integration-by-parts formula in Corollary 5 holds for this η12. Consequently, the following equality holds for all η12’s which are indicators of sets in H: (53)E[0Tη12(u)ψ*(u)du]=E[0Tη12(u)γ1(u)e2f(u)e2I{X(u-)=e1}du]. Since the set of all jump times of the chain X has “dt’’-measure zero, (54)E[0Tη12(u)ψ*(u)du]=E[0Tη12(u)γ1(u)e2f(u)e2I{X(u)=e1}du]. On the set {X(u)=e1}, (55)e2f(u)e2=e2(diag[A(u)e1]-diag[e1]A(u)-A(u)diag[e1]-diag[e1]A(u))e2=a21(u). Consequently, for all η12’s which are indicators of sets in H, (56)E[0Tη12(u)ψ*(u)du]=E[0Tη12(u)γ1(u)a21(u)I{X(u)=e1}du]. Note that

H generates the FX-predictable σ-field on the product space T×Ω;

the processes {γ1(u)uT} and {ψ*(u)uT} are FX-predictable.

Then (57)ψ*(u)=γ1(u)a21(u)I{X(u)=e1},for  almost  all(u,ω)T×Ω. Consequently, for almost all (u,ω)T×Ω, (58)E[x1G(J(T))FX(u-)]a12(u)I{X(u)=e1}=γ1(u)a21(u)I{X(u)=e1}. Then, (59)γ1(u)=E[x1G(J(T))FX(u-)]a12(u)a21(u), on the set {X(u)=e1}.

5. An Extension to a Function of Path Integrals

The integration-by-parts formulas and the martingale representation developed in the previous sections are now extended to a function of the integrals with respect to the whole paths of the fundamental jump processes relating to the chain X. This function may be considered a canonical form of an FX(T)-measurable random variable.

Consider an FX(T)-measurable random variable H which is of the following canonical form: (60)Hh(0Tη12(t)dJ12(t),0Tη21(t)dJ21(t)), where h:R2R is any measurable, integrable, and differentiable function. Note that H depends on the whole paths of the fundamental jump processes relating to the chain X; η12 and η21 are nonnegative, P-a.s. bounded, FX-predictable processes as defined in Section 3.

We now define some notation. Write (61)I12(T)0Tη12(t)dJ12(t),I21(T)0Tη21(t)dJ21(t),I(T)(I12(T),I21(T))R2. Then (62)H=h(I(T)).

The following theorem gives an extension to the integration-by-parts formula presented in Theorem 4 for the function h.

Theorem 14.

For each tT, let (63)λ~12(t)η12(t)λ12(t),λ~21(t)η21(t)λ21(t),φ~  (t)(0tλ~12(u)du,0tλ~21(u)du)R2. Then (64)E[Dxh(I(T)),φ~(T)]=E[h(I(T))0Tη(u)dM-(u)].

Proof.

The proof of this theorem resembles that of Theorem 4. We only give some key steps. For each ϵ>0, let (65)I12ϵ(T)0Tη12(t)dJ12ϵ(t),I21ϵ(T)0Tη21(t)dJ21ϵ(t). Write (66)Iϵ(T)(I12ϵ(T),I21ϵ(T))R2. By Lemma 2, the Pϵ-probability law of Iϵ(T) is the same as the P-law of I(T). Then (67)E[h(I(T))]=Eϵ[h(Iϵ(T))]=E[Λϵ(T)h(Iϵ(T))]. Differentiating with respect to ϵ and setting ϵ=0 give (68)E[ϵΛϵ(T)|ϵ=0h(Iϵ(T))|ϵ=0]+E[Λϵ(T)|ϵ=0Dxh(Iϵ(T)),ϵIϵ(T)|ϵ=0]=0. Then the result follows by noting that (69)ϵIϵ(T)|ϵ=0=(0Tλ~12(t)dt,0Tλ~21(t)dt)=φ~(T).

Similarly, the following corollaries are direct consequences of Theorem 14.

Corollary 15.

For any measurable, integrable, and differentiable function h:R2R, (70)E[x1h(I(T))0Tλ~12(u)du]=E[h(I(T))0Tη12(u)dM12(u)].

Corollary 16.

For any measurable, integrable, and differentiable function h:R2R, (71)E[x2h(I(T))0Tλ~21(u)du]=E[h(I(T))0Tη21(u)dM21(u)].

We now extend the martingale representation in Section 3 to the function Hh(I(T)) of the path integrals. By the martingale representation in Theorem 10, (72)h(I(T))=E[h(I(T))]+0Tγ~(u)dM-(u), for some FX-predictable process γ~{γ~(t)tT}.

Again by subtraction we assume that E[h(I(T))]=0. Then (73)h(I(T))=0Tγ~(u)dM-(u). The following theorem gives an expression for the integrand in the martingale representation for h(I(T)).

Theorem 17.

Suppose that a12(t),a21(t)>0 for each tT. Then the integrand γ~{γ~(t)tT}, where γ~(t)(γ~1(t),γ~2(t))R2, is determined by (74)γ~1(t)=E[x1h(I(T))FX(t-)]a12(t)a21(t),ontheset  {X(t)=e1},γ~2(t)=E[x2h(I(T))FX(t-)]a21(t)a12(t),ontheset  {X(t)=e2}.

Proof.

The proof resembles that of Theorem 13. We only need to note the fact that, for all η12’s which are indicators of sets in H, η122=η12.

6. An Application to Hedging Contingent Claims

In this section we will discuss an application of the martingale representation result derived in Section 4 to hedge contingent claims in the Markov chain financial market of Norberg . Here we consider a simplified version of the Markov chain market of Norberg , where there are two risky shares, namely, S1 and S2, and the Markov chain has only two states. We also suppose that the market interest rate is zero. In this case, as in Norberg , the (discounted) price processes of the two risky shares {S1(t)tT} and {S2(t)tT} under a risk-neutral probability, say P, are governed by (75)dSi(t)=Si(t-)((exp(β12i)-1)dM12(t)+(exp(β21i)-1)dM21(t)),Si(0)=si>0,i=1,2, where β12i and β21i, for i=1,2, are non-zero constants; {M12(t)tT} and {M21(t)tT} are (FX,P)-martingales. Note that the two risky shares are correlated since their price dynamics depend on M12 and M21.

For each i=1,2, let α12i=exp(β12i)-1 and let α21i=exp(β21i)-1. Then, as in Norberg , under the risk-neutral measure P, the (discounted) terminal prices S1(T) and S2(T) of the shares are given by (76)Si(T)=siexp(-α12i0Ta12(t)X(t-),e1dt-α21i0Ta21(t)X(t-),e2dt+β12iJ12(T)+β21iJ21(T)(-α12i0Ta12(t)X(t-),e1dt),  i=1,2. Consequently, the vector of the (discounted) terminal prices of the shares S(T)(S1(T),S2(T)) is a function of J(T)(J12(T),J21(T)).

We now consider a contingent claim H written on the two correlated risky shares S1 and S2 whose payoff at maturity T is a function of S(T), say H(S(T)). Two practical examples of contingent claims having payoffs of this form are an exchange option, which is also called a Margrabe option, and a quanto option.

Note that the payoffs of the Margrable option and the quanto option may not be differentiable functions of S(T). To apply the martingale representation result in Section 4 to derive the hedging quantities for the Margrable option and the quanto option, we need to consider approximations of H(S(T)) by some “smooth” or differentiable payoff functions of S(T). In the sequel, we suppose that, with a slight abuse of notation, H(S(T)) is such a “smooth” or differentiable payoff function of S(T).

Then, the payoff H(S(T)) can be written as (77)H(S(T))=G(J(T)), for some “suitable’’ measurable, differentiable and integrable function G:R2R.

Define, for each tT, a (2×2)-matrix Σ(t) by (78)Σ(t)(S1(t-)(exp(β121)-1)S1(t-)(exp(β211)-1)S2(t-)(exp(β122)-1)S2(t-)(exp(β212)-1))=(S1(t-)α121S1(t-)α211S2(t-)α122S2(t-)α212). Then the price processes of the two risky shares S1 and S2 under the risk-neutral measure P are governed by the following vector-valued stochastic differential equation: (79)dS(t)=Σ(t)dM-(t), where M-(t)(M12(t),M21(t)) as defined in Theorem 4.

Suppose α121α212α211α122. Then, the inverse Σ-1(t) of Σ(t) exists and is given by (80)Σ-1(t)=1α121α212-α211α122(α212S1(t-)-α211S2(t-)-α122S1(t-)α121S2(t-)). Consequently, (81)dM-(t)=Σ-1(t)dS(t).

By the martingale representation in Theorem 10, (82)H(S(T))=G(J(T))=E[G(J(T))]+0Tγ(u)dM-(u)=E[H(S(T))]+0Tγ(u)Σ-1(u)dS(u)=E[H(S(T))]+0T(γ1(u)α212-γ2(u)α122S1(u-))dS1(u)+0T(γ2(u)α121-γ1(u)α211S2(u-))dS2(u). Then the claim H(S(T)) can be hedged perfectly by constructing a dynamic portfolio which invests (γ1(t)α212-γ2(t)α122)/S1(t-) units of the risky share S1 and (γ2(t)α121-γ1(t)α211)/S2(t-) units of the risky share S2 at time t, for each tT. The initial investment of the portfolio is E[H(S(T))], which is the initial price of the claim H(S(T)). Using Theorem 13, γ1(t) and γ2(t) are determined as (83)γ1(t)=E[x1G(J(T))FX(t-)]a12(t)a21(t),on  the  set  {X(t)=e1},γ2(t)=E[x2G(J(T))FX(t-)]a21(t)a12(t),on  the  set  {X(t)=e2}.

We only illustrate here the use of the martingale representation result in Section 4 to hedge contingent claims whose payoffs depend only on the terminal prices of the risky shares in the Markov chain market. The martingale representation result in Section 5 may be used to hedge contingent claims with more general payoff structures in the Markov chain market.

7. Conclusion

An integration-by-parts formula for a function of the terminal values of the fundamental jump processes relating to a Markov chain was first established using the Bismut approach to Malliavin calculus. The formula was then applied to derive a new expression for the integrand in a stochastic integral in a martingale representation. The results were then extended to functions of the integrals with respect to the whole paths of the fundamental jump processes. These functions may be regarded as random variables of canonical forms. Only finite-dimensional calculus was needed in the derivations. Though some complex notations may be involved, the results presented here may be extended to the case of a general N-state Markov chain where a set of fundamental jump processes {Jik(t)tT}, i,k=1,2,,N, ik, is used. We applied the martingale representation result derived here to hedge a contingent claim written on two correlated risky shares in the Markov chain financial market of Norberg .

There are several future research directions based on the results developed in this paper which may be of theoretical and practical interests. The results may be applied to study the existence and uniqueness of densities of jump processes relating to a Markov chain. It seems that this problem is of fundamental importance in filtering and control theory of hidden Markov chains. Martingale representations play an important role in filtering and control. It may be interesting to explore the applications of the martingale representations developed in this paper in filtering and control for stochastic processes relating to Markov chains. The monograph by Elliott et al.  provided some discussions on the filtering and control of hidden Markov chains. The results developed here may be extended to develop Malliavin calculus for stochastic differential equations driven by a continuous-time, finite-state Markov chain and Markov regime-switching stochastic differential equations. It may be of practical interest to further explore the use of the martingale representation results developed here to hedge modern insurance products, such as unit-linked insurance products and longevity bonds in the Markov chain market of Norberg . In Bielecki et al. , the valuation of credit derivatives in a Markov chain model was discussed. It may be of practical interest to explore the application of the martingale representation results developed here to hedge credit derivatives in the Markov chain model discussed in Bielecki et al. .

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author would like to thank the editor and the reviewers for helpful comments.

Benth F. E. di Nunno G. Løkka A. Øksendal B. Proske F. Explicit representation of the minimal variance portfolio in markets driven by Lévy processes Mathematical Finance 2003 13 1 55 72 10.1111/1467-9965.t01-1-00005 MR1968096 ZBL1173.91377 León J. A. Navarro R. Nualart D. An anticipating calculus approach to the utility maximization of an insider Mathematical Finance 2003 13 1 171 185 10.1111/1467-9965.00012 ZBL1060.91054 Imkeller P. Malliavin's calculus in insider models: additional utility and free lunches Mathematical Finance 2003 13 1 153 169 10.1111/1467-9965.00011 MR1968102 ZBL1071.91017 Fournié E. Lasry J.-M. Lebuchoux J. Lions P.-L. Touzi N. Applications of Malliavin calculus to Monte Carlo methods in finance Finance and Stochastics 1999 3 4 391 412 10.1007/s007800050068 MR1842285 ZBL0947.60066 Fournié E. Lasry J.-M. Lebuchoux J. Lions P.-L. Applications of Malliavin calculus to Monte-Carlo methods in finance. II Finance and Stochastics 2001 5 2 201 236 10.1007/PL00013529 MR1841717 ZBL0973.60061 Bismut J. M. Martingales, the Malliavin calculus and hypoellipticity under general Hörmander's conditions Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 1981 56 4 469 505 10.1007/BF00531428 MR621660 ZBL0445.60049 Bichteler K. Gravereaux J.-B. Jacod J. Malliavin Calculus for Processes with Jumps 1987 2 New York, NY, USA Gordon and Breach Science Publishers MR1008471 Bass R. F. Cranston M. The Malliavin calculus for pure jump processes and applications to local time Annals of Probability 1986 14 490 532 10.1214/aop/1176992528 ZBL0595.60044 Norris J. Integration by Parts for Jump Processes Séminaire de Probabilités XXII 1998 1322 Berlin, Germany Springer Lecture Notes in Mathematics Elliott R. J. Tsoi A. H. Integration by parts for the single jump process Statistics & Probability Letters 1991 12 5 363 370 10.1016/0167-7152(91)90023-K MR1142088 ZBL0749.60044 Elliott R. J. Tsoi A. H. Integration by parts for Poisson processes Journal of Multivariate Analysis 1993 44 2 179 190 10.1006/jmva.1993.1010 MR1219202 ZBL0768.60052 Nualart D. The Malliavin Calculus and Related Topics 2006 2nd Berlin, Germany Springer MR2200233 Privault N. Stochastic Analysis in Discrete and Continuous Settings with Normal Martingales 2009 1982 Berlin, Germany Springer Lecture Notes in Mathematics 10.1007/978-3-642-02380-4 MR2531026 di Nunno G. Øksendal B. Proske F. Malliavin Calculus for Lévy Processes with Applications to Finance 2009 Berlin, Germany Springer Norberg R. A time-continuous markov chain interest model with applications to insurance Applied Stochastic Models and Data Analysis 1995 11 3 245 256 10.1002/asm.3150110306 ZBL1067.91509 Elliott R. J. Kopp P. E. Mathematics of Financial Markets 2005 2nd New York, NY, USA Springer MR2098795 Song N. Ching W. K. Siu T. K. Fung E. S. Ng M. K. Option valuation under a multivariate Markov chain model 1 Proceedings of the IEEE Computer Society Proceedings (CSO '10) Huangshan, China 177 181 Elliott R. J. Liew C. C. Siu T. K. Characteristic functions and option valuation in a Markov chain market Computers & Mathematics with Applications 2011 62 1 65 74 10.1016/j.camwa.2011.04.050 MR2821808 ZBL1228.91069 van der Hoek J. Elliott R. J. American option prices in a Markov chain market model Applied Stochastic Models in Business and Industry 2012 28 1 35 59 10.1002/asmb.893 MR2898900 ZBL06292430 van der Hoek J. Elliott R. J. Asset pricing using finite state Markov chain stochastic discount functions Stochastic Analysis and Applications 2012 30 5 865 894 10.1080/07362994.2012.704852 MR2966103 ZBL1258.91079 Norberg R. The Markov chain market ASTIN Bulletin 2003 33 265 287 10.2143/AST.33.2.503693 ZBL1098.91531 Koller M. Stochastic Models in Life Insurance 2012 Berlin, Germany Springer Tong H. Nonlinear Time Series: A Dynamical System Approach 1990 6 Oxford, UK Oxford University Press MR1079320 Shen H. Chu Y. Xu S. Zhang Z. Delay-dependent H control for jumping delayed systems with two Markov processes International Journal of Control, Automation, and Systems 2011 9 3 437 441 10.1007/s12555-011-0302-4 He S. Liu F. Finite-time H control of nonlinear jump systems with time-delays via dynamic observer-based state feedback IEEE Transactions on Fuzzy Systems 2012 20 4 605 614 10.1109/TFUZZ.2011.2177842 He S. Liu F. Adaptive observer-based fault estimation for stochastic Markovian jumping systems Abstract and Applied Analysis 2012 2012 11 176419 MR2947688 ZBL1246.93107 10.1155/2012/176419 Zhang X. Elliott R. J. Siu T. K. A stochastic maximum principle for a Markov regime-switching jump-diffusion model and its application to finance SIAM Journal on Control and Optimization 2012 50 2 964 990 10.1137/110839357 MR2914237 ZBL1244.93180 He S. Resilient L1-L filtering of uncertain Markovian jumping systems within the finite-time interval Abstract and Applied Analysis 2013 2013 7 791296 MR3044995 10.1155/2013/791296 Siu T. K. A BSDE approach to optimal investment of an insurer with hidden regime switching Stochastic Analysis and Applications 2013 31 1 1 18 10.1080/07362994.2012.727144 MR3007880 ZBL1267.91087 Elliott R. J. Siu T. K. Filtering and change point estimation for hidden Markov-modulated Poisson processes Applied Mathematics Letters 2014 28 66 71 10.1016/j.aml.2013.10.001 MR3128650 Wu Z.-G. Shi P. Su H. Chu J. Asynchronous l2-l filtering for discrete-time stochastic Markov jump systems with randomly occurred sensor nonlinearities Automatica 2014 50 1 180 186 10.1016/j.automatica.2013.09.041 MR3157739 Elliott R. J. Aggoun L. Moore J. B. Hidden Markov Models: Estimation and Control 1994 29 New York, NY, USA Springer MR1323178 Yin G. G. Zhang Q. Discrete-Time Markov Chains: Two-Time-Scale Methods and Applications 2005 55 New York, NY, USA Springer MR2092994 Yin G. G. Zhang Q. Continuous-Time Markov Chains and Applications: A Two-Time-Scale Approach 2013 37 2nd New York, NY, USA Springer 10.1007/978-1-4614-4346-9 MR2985157 Ching W. K. Huang X. Ng M. K. Siu T. K. Markov Chains: Models, Algorithms and Applications 2013 2nd New York, NY, USA Springer 10.1007/978-1-4614-6312-2 MR3053058 Elliott R. J. A partially observed control problem for Markov chains Applied Mathematics and Optimization 1992 25 2 151 169 10.1007/BF01182478 MR1142679 ZBL0762.93009 Elliott R. J. Stochastic Calculus and Applications 1982 18 Berlin, Germany Springer MR678919 Elliott R. J. Kohlmann M. Integration by parts and densities for jump processes Stochastics and Stochastics Reports 1989 27 2 83 97 MR1011656 10.1080/17442508908833569 ZBL0677.60058 Clark J. M. C. The representation of functionals of Brownian motion by stochastic integrals Annals of Mathematical Statistics 1970 41 1282 1295 MR0270448 10.1214/aoms/1177696903 ZBL0213.19402 Haussmann U. G. On the integral representation of functionals of Itô processes Stochastics 1979 3 1 17 28 10.1080/17442507908833134 MR546697 ZBL0427.60056 Ocone D. Malliavin's calculus and stochastic integral representations of functionals of diffusion processes Stochastics 1984 12 3-4 161 185 10.1080/17442508408833299 MR749372 ZBL0542.60055 Ocone D. L. Karatzas I. A generalized Clark representation formula, with application to optimal portfolios Stochastics and Stochastics Reports 1991 34 3-4 187 220 MR1124835 10.1080/17442509108833682 ZBL0727.60070 Karatzas I. Ocone D. L. Li J. An extension of Clark's formula Stochastics and Stochastics Reports 1991 37 3 127 131 MR1148344 10.1080/17442509108833731 ZBL0745.60056 Elliott R. J. Kohlmann M. A short proof of a martingale representation result Statistics & Probability Letters 1988 6 5 327 329 10.1016/0167-7152(88)90008-9 MR933291 ZBL0645.60053 Aase K. Øksendal B. Privault N. Ubøe J. White noise generalizations of the Clark-Haussmann-Ocone theorem with application to mathematical finance Finance and Stochastics 2000 4 4 465 496 10.1007/PL00013528 MR1779589 ZBL0963.60065 di Nunno G. Øksendal B. Proske F. White noise analysis for Lévy processes Journal of Functional Analysis 2004 206 1 109 148 10.1016/S0022-1236(03)00184-8 MR2024348 ZBL1078.60054 Brémaud P. Point Processes and Queues: Martingale Dynamics 1981 Berlin, Germany Springer MR636252 Bielecki T. R. Crepey S. Herbertsson A. Rennie A. Lipton A. Markov chain models of portfolio credit risk Oxford Handbook of Credit Derivatives 2011 Oxford, UK Oxford University Press 327 382