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Integration by Parts and Martingale Representation for a Markov Chain

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  • Tak Kuen Siu

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.

Suggested Citation

  • Tak Kuen Siu, 2014. "Integration by Parts and Martingale Representation for a Markov Chain," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    • Handle: RePEc:wly:jnlaaa:v:2014:y:2014:i:1:n:438258
    DOI: 10.1155/2014/438258
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    References listed on IDEAS

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    1. Elliott, R. J. & Tsoi, A. H., 1993. "Integration by Parts for Poisson Processes," Journal of Multivariate Analysis, Elsevier, vol. 44(2), pages 179-190, February.
    2. Shuping He & Fei Liu, 2012. "Adaptive Observer‐Based Fault Estimation for Stochastic Markovian Jumping Systems," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    3. Eric Fournié & Jean-Michel Lasry & Pierre-Louis Lions & Jérôme Lebuchoux & Nizar Touzi, 1999. "Applications of Malliavin calculus to Monte Carlo methods in finance," Finance and Stochastics, Springer, vol. 3(4), pages 391-412.
    4. Jorge A. León & Reyla Navarro & David Nualart, 2003. "An Anticipating Calculus Approach to the Utility Maximization of an Insider," Mathematical Finance, Wiley Blackwell, vol. 13(1), pages 171-185, January.
    5. Shuping He & Fei Liu, 2012. "Adaptive Observer-Based Fault Estimation for Stochastic Markovian Jumping Systems," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-11, July.
    6. Fred Espen Benth & Giulia Di Nunno & Arne Løkka & Bernt Øksendal & Frank Proske, 2003. "Explicit Representation of the Minimal Variance Portfolio in Markets Driven by Lévy Processes," Mathematical Finance, Wiley Blackwell, vol. 13(1), pages 55-72, January.
    7. Peter Imkeller, 2003. "Malliavin's Calculus in Insider Models: Additional Utility and Free Lunches," Mathematical Finance, Wiley Blackwell, vol. 13(1), pages 153-169, January.
    8. Elliott, Robert J. & Tsoi, Allanus H., 1991. "Integration by parts for the single jump process," Statistics & Probability Letters, Elsevier, vol. 12(5), pages 363-370, November.
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    Cited by:

    1. Zheng Gu & Yue Liu & Aijun Yang & Kaodui Li, 2022. "New Method of Sensitivity Computation Based on Markov Models with Its Application for Risk Management," Journal of Mathematics, John Wiley & Sons, vol. 2022(1).

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