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 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.

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

“…✷ Remark 2.2. We remark that the well-posedness of SPDE (1) is also valid for non-Lipschitz assumptions on f possibly depending on the spatial variable, which was proposed in [4,5]. In particular, we may assume that there exists positive constants L 1 < γ and L 2 such that for any x ∈ O and any u, v ∈ R,…”

“…Elliptic SPDEs driven by white noises and colored noises have been considered by many authors, see e.g. [1,5,6,15,16] for white noises, [11,12,15] for colored noises determined by Riesz-type kernels, [3,4] for fractional noises, and [14] for power-law noises.…”

Numerical Approximation for Stochastic Nonlinear Fractional Diffusion Equation Driven by Rough Noise

“…But for the case H ∈ (0, 1 2 ) (called as "rough noise"), the existing discussions seem to be few. In [10], the authors propose numerical analyses about the second-order stochastic differential equation driven by spatial fractional Gaussian noise with H ∈ (0, 1 2 ); the reference [23] uses the equivalence of different fractional Sobolev spaces and the assumption τ < τ * (τ * depends on the spatial discretization) to provide a unified strong convergence analysis for fractional stochastic partial differential equation driven by fractional cylinder noise with H ∈ (0, 1); in [9], the authors propose the regularity estimates and the corresponding numerical analyses about the stochastic evolution equation driven by fractional Brownian sheet with H 1 ∈ (0, 1 2 ) and H 2 = 1 2 . In this paper, we focus on the fractional diffusion equation driven by fractional Brownian sheet with Hurst parameters H 1 , H 2 ∈ (0, 1 2 ] numerically, which are both rough in the temporal and spatial directions.…”

“…Then the Wong-Zakai approximation [16,17] is used to regularize the fractional Brownian sheet noise ξ H1,H2 . The existing discussions [9,10,21] rely on the Green function of Eq. ( 1) composed of Mittag-Leffler function [27], making the convergence analysis of the Wong-Zakai approximation complicated.…”

A unified convergence analysis for the fractional diffusion equation driven by fractional Gaussion noise with Hurst index $H\in(0,1)$

“…In the past few decades, there have been some numerical discussions for the stochastic PDEs driven by fractional Gaussian noise with the index H ∈ (1/2, 1) or H = 1/2 [3,15,16,25,26,27]. In addition, [6,7] use the finite element method to solve the PDE driven by spatial fractional Gaussian noise with an index H ∈ (0, 1/2), where some special Green functions and Itô isometry are used to provide the regularity of the solution, but these techniques can not reflect the influence of the temporal fractional Gaussian noise with H ∈ (0, 1/2) on the regularity of the mild solution and there are hardly researches for the temporal fractional Gaussian noise with H ∈ (0, 1/2). To try to fill the gap, a unified argument for H ∈ (0, 1) is proposed in this paper.…”