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

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 ∂ t 2H = 2Ht 2H−1 ∂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.