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On decomposition of the last passage time of diffusions

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  • Egami, Masahiko
  • Kevkhishvili, Rusudan

Abstract

For a regular transient diffusion, we derive the decomposition formula of the Laplace transform of the last passage time to a certain state α explicitly in a simple form in terms of the Green functions, which also leads to the Green function’s decomposition formula. This is accomplished by transforming the original diffusion into two diffusions using the occupation time of the area above and below α. We demonstrate applications of the decomposition formulas to various diffusions including a Brownian motion with two-valued drift and present a financial example of the leverage effect caused by the stock price with switching volatility.

Suggested Citation

  • Egami, Masahiko & Kevkhishvili, Rusudan, 2025. "On decomposition of the last passage time of diffusions," Stochastic Processes and their Applications, Elsevier, vol. 182(C).
    • Handle: RePEc:eee:spapps:v:182:y:2025:i:c:s030441492500002x
    DOI: 10.1016/j.spa.2025.104563
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    References listed on IDEAS

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    1. Masahiko Egami & Rusudan Kevkhishvili, 2020. "Time reversal and last passage time of diffusions with applications to credit risk management," Finance and Stochastics, Springer, vol. 24(3), pages 795-825, July.
    2. Bae, Jinho & Kim, Chang-Jin & Nelson, Charles R., 2007. "Why are stock returns and volatility negatively correlated?," Journal of Empirical Finance, Elsevier, vol. 14(1), pages 41-58, January.
    3. Dayanik, Savas & Karatzas, Ioannis, 2003. "On the optimal stopping problem for one-dimensional diffusions," Stochastic Processes and their Applications, Elsevier, vol. 107(2), pages 173-212, October.
    4. Ashkan Nikeghbali & Eckhard Platen, 2013. "A reading guide for last passage times with financial applications in view," Finance and Stochastics, Springer, vol. 17(3), pages 615-640, July.
    5. R. J. Elliott & M. Jeanblanc & M. Yor, 2000. "On Models of Default Risk," Mathematical Finance, Wiley Blackwell, vol. 10(2), pages 179-195, April.
    6. Patrick Cheridito & Ashkan Nikeghbali & Eckhard Platen, 2012. "Processes of Class Sigma, Last Passage Times, and Drawdowns," Published Paper Series 2012-4, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
    7. Enrico Bibbona & Susanne Ditlevsen, 2013. "Estimation in Discretely Observed Diffusions Killed at a Threshold," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 40(2), pages 274-293, June.
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