Numerical methods for the two-dimensional Fokker-Planck equation governing the probability density function of the tempered fractional Brownian motion

TL;DR

This paper develops and analyzes numerical schemes for solving the two-dimensional Fokker-Planck equation related to tempered fractional Brownian motion, addressing singularities and improving computational efficiency for different Hurst parameter ranges.

Contribution

It introduces novel numerical methods with nonuniform time discretization to effectively solve the Fokker-Planck equation for tempered fractional Brownian motion.

Findings

Numerical schemes successfully compute the probability density function.

Mean squared displacement matches theoretical properties.

Enhanced efficiency for different Hurst parameters.

Abstract

In this paper, we study the numerical schemes for the two-dimensional Fokker-Planck equation governing the probability density function of the tempered fractional Brownian motion. The main challenges of the numerical schemes come from the singularity in the time direction. When $0<H<0.5$, a change of variables $\partial \left(t^{2H}\right)=2Ht^{2H-1}\partial t$ avoids the singularity of numerical computation at $t=0$, which naturally results in nonuniform time discretization and greatly improves the computational efficiency. For $0.5<H<1$, the time span dependent numerical scheme and nonuniform time discretization are introduced to ensure the effectiveness of the calculation and the computational efficiency. By numerically solving the corresponding Fokker-Planck equation, we obtain the mean squared displacement of stochastic processes, which conforms to the characteristics of the tempered fractional Brownian motion.