Dynamical behavior of a nonlocal Fokker-Planck equation for a stochastic system with tempered stable noise
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.
Keywords
Cite
@article{arxiv.2103.16834,
title = {Dynamical behavior of a nonlocal Fokker-Planck equation for a stochastic system with tempered stable noise},
author = {Li Lin and Jinqiao Duan and Xiao Wang and Yanjie Zhang},
journal= {arXiv preprint arXiv:2103.16834},
year = {2021}
}