Fractional Langevin Equation Driven by Multifractional Brownian Motion: Integral Equation Approach

Abstract

The fractional Langevin stochastic differential equation driven by multifractional Brownian motion of Riemann-Liouville type, which describes the long-range interactions and its mean square displacement has a power-law growth in time $〈x^2(t)〉≃t^α,$ where $0<α<1$ correspond to the subdiffusion, and α is the fractional order. In this paper, we extend the framework to account for time-varying environmental properties, leading to a variable-order fractional Langevin equation. We give the Euler-Maruyama scheme of the solution and then prove the strong convergence of the Euler-Maruyama scheme. Numerical experiments with time-dependent Hurst indices are presented to illustrate the theoretical findings.