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The fundamental solution of the master equation for a jump‐diffusion Ornstein–Uhlenbeck process

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  • Olga S. Rozanova
  • Nikolai A. Krutov

Abstract

An integro‐differential equation for the probability density of the generalized stochastic Ornstein–Uhlenbeck process with jump diffusion is considered for a special case of the Laplacian distribution of jumps. It is shown that for a certain ratio between the intensity of jumps and the speed of reversion, the fundamental solution can be found explicitly, as a finite sum. Alternatively, the fundamental solution can be represented as converging power series. The properties of this solution are investigated. The fundamental solution makes it possible to obtain explicit formulas for the density at each instant of time, which is important, for example, for testing numerical methods.

Suggested Citation

  • Olga S. Rozanova & Nikolai A. Krutov, 2024. "The fundamental solution of the master equation for a jump‐diffusion Ornstein–Uhlenbeck process," Mathematische Nachrichten, Wiley Blackwell, vol. 297(8), pages 3052-3063, August.
    • Handle: RePEc:bla:mathna:v:297:y:2024:i:8:p:3052-3063
    DOI: 10.1002/mana.202300200
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    References listed on IDEAS

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    1. Gaviraghi, B. & Annunziato, M. & Borzì, A., 2017. "Analysis of splitting methods for solving a partial integro-differential Fokker–Planck equation," Applied Mathematics and Computation, Elsevier, vol. 294(C), pages 1-17.
    2. Nicholas F Marko & Robert J Weil, 2012. "Non-Gaussian Distributions Affect Identification of Expression Patterns, Functional Annotation, and Prospective Classification in Human Cancer Genomes," PLOS ONE, Public Library of Science, vol. 7(10), pages 1-15, October.
    3. S. I. Denisov & W. Horsthemke & P. Hänggi, 2009. "Generalized Fokker-Planck equation: Derivation and exact solutions," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 68(4), pages 567-575, April.
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