Numerical approximation for a stochastic Caputo fractional differential equation with multiplicative noise

Abstract

We investigate a numerical method for approximating stochastic Caputo fractional differential equations driven by multiplicative noise. The nonlinear functions f and g are assumed to satisfy the global Lipschitz conditions as well as the linear growth conditions. The noise is approximated by a piecewise constant function, yielding a regularized stochastic fractional differential equation. We prove that the error between the exact solution and the solution of the regularized equation converges in the L2((0,T)×Ω) norm with an order of O(Δtα−1/2), where α∈(1/2,1] is the order of the Caputo fractional derivative, and Δt is the time step size. Numerical experiments are provided to confirm that the simulation results are consistent with the theoretical convergence order.