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2020-02

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cc-by-nc-nd (c) Elsevier B.V., 2020
Please use this identifier to cite or link to this item: https://hdl.handle.net/2445/193773

Dyson type formula for pure jump Lévy processes with some applications to finance

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https://doi.org/10.1016/j.spa.2019.03.019

Abstract

In this paper we obtain a Dyson type formula for integrable functionals of a pure jump Lévy process. We represent the conditional expectation of a $\mathscr{F}_T$-measurable random variable $F$ at a time $t \leq T$ as an exponential formula involving Malliavin derivatives evaluated along a frozen path. The series representation of this exponential formula turns out to be useful for different applications, and in particular in quantitative finance, as we show in the following examples: the first one is the pricing of options in the Poisson-Black-Scholes model; the second one is the pricing of discount bonds in the Lévy quadratic model. We also obtain, for the conditional expectation of a function of a finite number of the process values, a backward Taylor expansion, that turns out to be useful for numerical calculations.

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Equacions en derivades parcials, Processos estocàstics, Teoria de jocs, Distribució (Teoria de la probabilitat)

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Partial differential equations, Stochastic processes, Game theory, Distribution (Probability theory)

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Articles publicats en revistes (Matemàtiques i Informàtica)

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JIN, Sixian, SCHELLHORN, Henry and VIVES I SANTA EULÀLIA, Josep. Dyson type formula for pure jump Lévy processes with some applications to finance. Stochastic Processes and their Applications. 2020. Vol. 130, num. 2, pags. 824-844. ISSN 0304-4149. [consulted: 15 of August of 2026]. Available at: https://hdl.handle.net/2445/193773

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