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Occupied Processes: Going with the Flow

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  • Valentin Tissot-Daguette

Abstract

We develop an It\^o calculus for functionals of the "time" spent by a path at arbitrary levels. A Markovian setting is recovered by lifting a process $X$ with its flow of occupation measures $\mathcal{O}$ and call the pair $(\mathcal{O},X)$ the occupied process. While the occupation measure erases the chronology of the path, we show that our framework still includes many relevant problems in stochastic analysis and financial mathematics. The study of occupied processes therefore strikes a middle ground between the path-independent case and Dupire's Functional It\^o Calculus. We extend It\^o's and Feynman-Kac's formula by introducing the occupation derivative, a projection of the functional linear derivative used extensively in mean field games and McKean-Vlasov optimal control. Importantly, we can recast through Feynman-Kac's theorem a large class of path-dependent PDEs as parabolic problems where the occupation measure plays the role of time. We apply the present tools to the optimal stopping of spot local time and discuss financial examples including exotic options, corridor variance swaps, and path-dependent volatility.

Suggested Citation

  • Valentin Tissot-Daguette, 2023. "Occupied Processes: Going with the Flow," Papers 2311.07936, arXiv.org, revised Dec 2023.
  • Handle: RePEc:arx:papers:2311.07936
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    1. Christa Cuchiero & Wahid Khosrawi & Josef Teichmann, 2020. "A generative adversarial network approach to calibration of local stochastic volatility models," Papers 2005.02505, arXiv.org, revised Sep 2020.
    2. Christa Cuchiero & Francesco Guida & Luca di Persio & Sara Svaluto-Ferro, 2021. "Measure-valued affine and polynomial diffusions," Papers 2112.15129, arXiv.org.
    3. Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," The Journal of Business, University of Chicago Press, vol. 51(4), pages 621-651, October.
    4. Julien-N. Hugonnier, 1999. "The Feynman–Kac Formula And Pricing Occupation Time Derivatives," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 2(02), pages 153-178.
    5. David G. Hobson & L. C. G. Rogers, 1998. "Complete Models with Stochastic Volatility," Mathematical Finance, Wiley Blackwell, vol. 8(1), pages 27-48, January.
    6. Julien Guyon & Jordan Lekeufack, 2023. "Volatility is (mostly) path-dependent," Quantitative Finance, Taylor & Francis Journals, vol. 23(9), pages 1221-1258, September.
    7. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," University of California at Los Angeles, Anderson Graduate School of Management qt43n1k4jb, Anderson Graduate School of Management, UCLA.
    8. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," The Review of Financial Studies, Society for Financial Studies, vol. 14(1), pages 113-147.
    9. Dawson, D. A., 1975. "Stochastic evolution equations and related measure processes," Journal of Multivariate Analysis, Elsevier, vol. 5(1), pages 1-52, March.
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