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Abstract and Applied Analysis
Volume 2014, Article ID 438258, 11 pages
http://dx.doi.org/10.1155/2014/438258
Research Article

Integration by Parts and Martingale Representation for a Markov Chain

1Cass Business School, City University London, 106 Bunhill Row, London EC1Y 8TZ, UK
2Department of Applied Finance and Actuarial Studies, Faculty of Business and Economics, Macquarie University, Sydney, NSW 2109, Australia

Received 30 October 2013; Accepted 10 May 2014; Published 2 June 2014

Academic Editor: Shuping He

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.

Abstract

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.