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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.

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. [

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 [

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 [

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 [

The rest of the paper is organized as follows. Section

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

To specify the probability laws of the chain

Let

For each

Define, for each

The following lemma gives the semimartingale decomposition for

For each

The proof of this lemma is standard. For the sake of completeness, we present the proof here:

From Lemma

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 each

Define, for each

For each

Define, for each

Then, for each

A new probability measure

The

By a version of Girsanov’s theorem, the process

The

To simplify our notation and illustrate the main idea, we consider the situation where the chain

Let

Write

Define the following gradient of

For each

By a version of Bayes' rule,

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

For any measurable, integrable, and differentiable function

The result follows by putting

For any measurable, integrable, and differentiable function

The result follows by putting

The integration-by-parts formula in Corollary

In Elliott and Kohlmann [

In the Markov chain financial market of Norberg [

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 [

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 [

Note that the filtration

For any real-valued, square-integrable

Furthermore, we need the following expression for the predictable quadratic variation

Let

To simplify our notation, let

Then

For each

Recall that

By the martingale representation presented in Theorem

It can be supposed that

The integrand

Suppose that

We only give the proof for the integrand

We now take

the processes

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

Consider an

We now define some notation. Write

The following theorem gives an extension to the integration-by-parts formula presented in Theorem

For each

The proof of this theorem resembles that of Theorem

Similarly, the following corollaries are direct consequences of Theorem

For any measurable, integrable, and differentiable function

For any measurable, integrable, and differentiable function

We now extend the martingale representation in Section

Again by subtraction we assume that

Suppose that

The proof resembles that of Theorem

In this section we will discuss an application of the martingale representation result derived in Section

For each

We now consider a contingent claim

Note that the payoffs of the Margrable option and the quanto option may not be differentiable functions of

Then, the payoff

Define, for each

Suppose

By the martingale representation in Theorem

We only illustrate here the use of the martingale representation result in Section

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

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. [

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

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