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.
In this paper, the option hedging problem for a Markov-modulated exponential Lévy model is examined. We use the local risk-minimization approach to study optimal hedging strategies for Europeans derivatives when the price of the underlying is given by a regime-switching Lévy model. We use a martingale representation theorem result to construct an explicit local risk minimizing strategy.
It is known that the market in a Markovian regime-switching model is, in general, incomplete, so not all contingent claims can be perfectly hedged. We show, in this paper, how certain contingent claims are attainable in the regime-switching market using a money market account, a share and a zero-coupon bond. General contingent claims with payoffs depending on both the share price and the state of the regime-switching process are considered. We apply a martingale representation result to show the attainability of a European-style contingent claim. We also extend our analysis to Asian-style and American-style contingent claims.
We introduce a new approach to simulating rare events for Markov random walks with heavy-tailed increments. This approach involves sequential importance sampling and resampling, and uses a martingale representation of the corresponding estimate of the rare-event probability to show that it is unbiased and to bound its variance. By choosing the importance measures and resampling weights suitably, it is shown how this approach can yield asymptotically efficient Monte Carlo estimates.
Martingale representation processes and applications in the market viability under information flow expansion
