Martingale Representation and All That

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.

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Martingale representation theorems for initially enlarged filtrations

Stochastic Processes and their Applications ◽

10.1016/s0304-4149(00)00015-6 ◽

2000 ◽

Vol 89 (1) ◽

pp. 101-116 ◽

Cited By ~ 44

Author(s):

Jürgen Amendinger

Keyword(s):

Representation Theorems ◽

Martingale Representation

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LOCAL RISK-MINIMIZATION UNDER MARKOV-MODULATED EXPONENTIAL LÉVY MODEL

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024915500338 ◽

2015 ◽

Vol 18 (05) ◽

pp. 1550033

Author(s):

OLIVIER MENOUKEU-PAMEN ◽

ROMUALD MOMEYA

Keyword(s):

Regime Switching ◽

Risk Minimization ◽

Martingale Representation ◽

Option Hedging ◽

Hedging Strategies ◽

Local Risk ◽

Hedging Problem ◽

Markov Modulated ◽

Martingale Representation Theorem ◽

Minimization Approach

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.

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ATTAINABLE CONTINGENT CLAIMS IN A MARKOVIAN REGIME-SWITCHING MARKET

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024912500550 ◽

2012 ◽

Vol 15 (08) ◽

pp. 1250055 ◽

Cited By ~ 3

Author(s):

ROBERT J. ELLIOTT ◽

TAK KUEN SIU

Keyword(s):

Regime Switching ◽

Contingent Claims ◽

Share Price ◽

Regime Switching Model ◽

Martingale Representation ◽

Switching Model ◽

Representation Result ◽

Markovian Regime Switching ◽

Coupon Bond ◽

Markovian Regime

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.

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Rare-Event Simulation of Heavy-Tailed Random Walks by Sequential Importance Sampling and Resampling

Advances in Applied Probability ◽

10.1017/s000186780000608x ◽

2012 ◽

Vol 44 (04) ◽

pp. 1173-1196

Author(s):

Hock Peng Chan ◽

Shaojie Deng ◽

Tze-Leung Lai

Keyword(s):

Random Walks ◽

Importance Sampling ◽

Rare Event ◽

Sequential Importance Sampling ◽

New Approach ◽

Martingale Representation ◽

Event Simulation ◽

Markov Random ◽

Heavy Tailed ◽

Asymptotically Efficient

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.

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