Dynamical behavior of a nonlocal Fokker-Planck equation for a stochastic system with tempered stable noise

TL;DR

This paper studies a stochastic system with tempered stable noise, proving existence and uniqueness of solutions to its nonlocal Fokker-Planck equation, and develops a numerical method for simulation, with applications to nonlinear filtering.

Contribution

It establishes a superposition principle linking solutions to the nonlocal Fokker-Planck equation with martingale solutions, and introduces a convergent finite difference scheme for simulation.

Findings

Proved existence and uniqueness of strong solutions.

Developed a convergent finite difference numerical method.

Applied results to simulate a nonlocal Zakai equation.

Abstract

We characterize a stochastic dynamical system with tempered stable noise, by examining its probability density evolution. This probability density function satisfies a nonlocal Fokker-Planck equation. First, we prove a superposition principle that the probability measure-valued solution to this nonlocal Fokker-Planck equation is equivalent to the martingale solution composed with the inverse stochastic flow. This result together with a Schauder estimate leads to the existence and uniqueness of strong solution for the nonlocal Fokker-Planck equation. Second, we devise a convergent finite difference method to simulate the probability density function by solving the nonlocal Fokker-Planck equation. Finally, we apply our aforementioned theoretical and numerical results to a nonlinear filtering system by simulating a nonlocal Zakai equation.