Numerical analysis and applications of Fokker-Planck equations for stochastic dynamical systems with multiplicative α-stable noises

摘要

In this paper, we study the nonlocal Fokker-Planck equations (FPEs) associated with Levy driven scalar stochastic dynamical systems. We first derive the Fokker-Planck equation for the case of multiplicative symmetric α-stable noises, by the adjoint operator method. Then we construct a finite difference scheme to simulate the nonlocal FPE on either bounded or infinite domain. It is shown that the semi-discrete scheme satisfies the discrete maximum principle and converges. Some experiments are conducted to validate the numerical method. Finally, we extend the results to the asymmetric case and present an application to the nonlinear filtering problem.