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Kac's moment formula and the Feynman-Kac formula for additive functionals of a Markov process

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  • Fitzsimmons, P. J.
  • Pitman, Jim

Abstract

Mark Kac introduced a method for calculating the distribution of the integral Av=[integral operator]0Tv(Xt) dt for a function v of a Markov process (Xt, t[greater-or-equal, slanted]0) and a suitable random time T, which yields the Feynman-Kac formula for the moment-generating function of Av. We review Kac's method, with emphasis on an aspect often overlooked. This is Kac's formula for moments of Av, which may be stated as follows. For any random time T such that the killed process (Xt, 0[less-than-or-equals, slant]t

Suggested Citation

  • Fitzsimmons, P. J. & Pitman, Jim, 1999. "Kac's moment formula and the Feynman-Kac formula for additive functionals of a Markov process," Stochastic Processes and their Applications, Elsevier, vol. 79(1), pages 117-134, January.
    • Handle: RePEc:eee:spapps:v:79:y:1999:i:1:p:117-134
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    References listed on IDEAS

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    1. Marcus, Michael B. & Rosen, Jay, 1995. "Logarithmic averages for the local times of recurrent random walks and Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 59(2), pages 175-184, October.
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    Cited by:

    1. Dell'Era Mario, M.D., 2008. "Pricing of Double Barrier Options by Spectral Theory," MPRA Paper 17502, University Library of Munich, Germany.
    2. Jackson Loper, 2020. "Uniform Ergodicity for Brownian Motion in a Bounded Convex Set," Journal of Theoretical Probability, Springer, vol. 33(1), pages 22-35, March.
    3. Dell'Era Mario, M.D., 2008. "Pricing of the European Options by Spectral Theory," MPRA Paper 17429, University Library of Munich, Germany.
    4. Alistair N Boettiger & Peter L Ralph & Steven N Evans, 2011. "Transcriptional Regulation: Effects of Promoter Proximal Pausing on Speed, Synchrony and Reliability," PLOS Computational Biology, Public Library of Science, vol. 7(5), pages 1-14, May.
    5. Chen, Xia, 2001. "Moderate deviations for Markovian occupation times," Stochastic Processes and their Applications, Elsevier, vol. 94(1), pages 51-70, July.
    6. Depperschmidt, Andrej & Pfaffelhuber, Peter, 2010. "Asymptotics of a Brownian ratchet for protein translocation," Stochastic Processes and their Applications, Elsevier, vol. 120(6), pages 901-925, June.
    7. Masaaki Fukasawa, 2010. "Asymptotic analysis for stochastic volatility: Edgeworth expansion," Papers 1004.2106, arXiv.org.
    8. Masaaki Fukasawa, 2010. "Central limit theorem for the realized volatility based on tick time sampling," Finance and Stochastics, Springer, vol. 14(2), pages 209-233, April.

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