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

TL;DR

This paper develops finite element methods for solving a one-dimensional second order stochastic differential equation driven by fractional Brownian motion with Hurst index up to 1/2, providing error estimates through convergence analysis.

Contribution

It introduces a novel finite element approximation scheme for equations driven by fractional Brownian motion with Hurst index ≤ 1/2, including rigorous error analysis.

Findings

Finite element approximations converge to the true solution.

Error estimates are derived for the approximation scheme.

The method effectively handles fractional noise with Hurst index up to 1/2.

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 con- stant approximations to construct the finite element approximations for the equation. The error estimate of the approximations is derived through rigorous convergence analysis.