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Local time-space stochastic calculus for Lévy processes

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  • Eisenbaum, Nathalie

Abstract

We develop a stochastic calculus on the plane with respect to the local times of a large class of Lévy processes. We can then extend to these Lévy processes an Itô formula that was established previously for Brownian motion. Our method provides also a multidimensional version of the formula. We show that this formula generates many "Itô formulas" that fit various problems. In the special case of a linear Brownian motion, we recover a recently established Itô formula that involves local times on curves. This formula is already used in financial mathematics.

Suggested Citation

  • Eisenbaum, Nathalie, 2006. "Local time-space stochastic calculus for Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 116(5), pages 757-778, May.
    • Handle: RePEc:eee:spapps:v:116:y:2006:i:5:p:757-778
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    References listed on IDEAS

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    1. Goran Peskir, 2005. "On The American Option Problem," Mathematical Finance, Wiley Blackwell, vol. 15(1), pages 169-181, January.
    2. Duistermaat, J.J. & Kyprianou, A.E. & van Schaik, K., 2005. "Finite expiry Russian options," Stochastic Processes and their Applications, Elsevier, vol. 115(4), pages 609-638, April.
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    Cited by:

    1. Likibi Pellat, Rhoss & Menoukeu Pamen, Olivier, 2024. "Density analysis for coupled forward–backward SDEs with non-Lipschitz drifts and applications," Stochastic Processes and their Applications, Elsevier, vol. 173(C).
    2. Cheng Cai & Tiziano De Angelis, 2021. "A change of variable formula with applications to multi-dimensional optimal stopping problems," Papers 2104.05835, arXiv.org, revised Jul 2023.
    3. Walsh, Alexander, 2011. "Local time-space calculus for symmetric Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 121(9), pages 1982-2013, September.
    4. Yang, Xiangfeng & Yan, Litan, 2007. "Some remarks on local time-space calculus," Statistics & Probability Letters, Elsevier, vol. 77(16), pages 1600-1607, October.
    5. Olivier Menoukeu-Pamen & Ludovic Tangpi, 2023. "Maximum Principle for Stochastic Control of SDEs with Measurable Drifts," Journal of Optimization Theory and Applications, Springer, vol. 197(3), pages 1195-1228, June.

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