Finite element approximations for second-order stochastic differential equation driven by fractional Brownian motion

Abstract

We consider finite element approximations for a one-dimensional second-order stochastic differential equation of boundary value type driven by a fractional Brownian motion with Hurst index |$H\,{\le}\, 1/2$|⁠. We make use of a sequence of approximate solutions with the fractional noise replaced by its piecewise constant approximations to construct the finite element approximations for the equation. The error estimate of the approximations is derived through rigorous convergence analysis.