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Numerical methods for the two-dimensional Fokker-Planck equation governing the probability density function of the tempered fractional Brownian motion. (English) Zbl 07235959
Summary: 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 \(\partial (t^{2H} )=2Ht^{2H-1}\partial t\) avoids the singularity of numerical computation at \(t = 0\), which naturally results in nonuniform time discretization and greatly improves the computational efficiency. For \(H > 0.5\), the time span dependent numerical scheme and nonuniform time discretization are introduced to ensure the effectiveness of the calculation and the computational efficiency. The stability and convergence of the numerical schemes are demonstrated by using Fourier method. 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.
MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
60J65 Brownian motion
60G22 Fractional processes, including fractional Brownian motion
35Q84 Fokker-Planck equations
Software:
FODE
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