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

• Derived the explicit Fokker-Planck equations for the SDE with multiplicative α stable noise. • Constructed a new numerical scheme for solving nonlocal Fokker-Planck equation. • Applied the results to a nonlinear filtering problem. • Results are expected to assist simulation study of, for example, the stochastic climate dynamics. In this paper, we study the nonlocal Fokker-Planck equations (FPEs) associated with Lévy-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. [ABSTRACT FROM AUTHOR]