Fokker-Planck equations for Marcus stochastic differential equations driven by Levy processes

TL;DR

This paper derives explicit Fokker-Planck equations for Marcus stochastic differential equations driven by Levy processes, extending known results from Gaussian to non-Gaussian noise in arbitrary dimensions.

Contribution

It provides a simple, general formula for the Fokker-Planck equations of Marcus SDEs driven by Levy processes, addressing an open problem in the field.

Findings

Derived explicit Fokker-Planck equations for Levy-driven Marcus SDEs

Presented examples demonstrating application of the formulas

Facilitated theoretical analysis and numerical computation of these equations

Abstract

Marcus stochastic differential equations (SDEs) often are appropriate models for stochastic dynamical systems driven by non-Gaussian Levy processes and have wide applications in engineering and physical sciences. The probability density of the solution to an SDE offers complete statistical information on the underlying stochastic process. Explicit formula for the Fokker-Planck equation, the governing equation for the probability density, is well-known when the SDE is driven by a Brownian motion. In this paper, we address the open question of finding the Fokker-Plank equations for Marcus SDEs in arbitrary dimensions driven by non-Gaussian Levy processes. The equations are given in a simple form that facilitates theoretical analysis and numerical computation. Several examples are presented to illustrate how the theoretical results can be applied to obtain Fokker-Planck equations for Marcus SDEs driven by Levy processes.