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

To appear in IMA Journal of Numerical Analysis; the time-dependent case such as stochastic heat equation and stochastic wave equation driven by fractional Brownian sheet with temporal Hurst index $1/2$ and spatial Hurst index $H\le 1/2$ has been considered by arXiv:1601.02085 for spatially Galerkin approximations and a forthcoming paper for fully discrete approximations

References in corpus (1)

• Well-posedness and Finite Element Approximations for Elliptic SPDEs with Gaussian Noises

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• Optimal Hölder Continuity and Hitting Probabilities for SPDEs with Rough Fractional Noises
• An inverse random source problem for the time-space fractional diffusion equation driven by fractional Brownian motion
• A unified convergence analysis for the fractional diffusion equation driven by fractional Gaussion noise with Hurst index $H\in (0,1)$