[Paper Review] Finite Element Approximations for Semilinear Elliptic SPDEs driven by Fractional Brownian Motion with Hurst Parameter $H<\frac{1}{2}$

Yanzhao Cao, Jialin Hong+1 more|arXiv (Cornell University)|Jul 9, 2015

Stochastic processes and financial applications18 references3 citations

TL;DR

This paper proposes a finite element method for solving one-dimensional semilinear elliptic SPDEs driven by fractional Brownian motion with Hurst parameter H < 1/2. By replacing the fractional noise with piecewise constant approximations and leveraging rigorous convergence analysis, the authors establish error estimates for the resulting numerical scheme, providing a well-posed and convergent approximation framework for this class of irregular SPDEs.

ABSTRACT

We consider finite element approximations for one dimensional boundary value problems of semilinear elliptic stochastic partial differential equations (SPDEs) driven by a fractional Brownian motion with Hurst parameter $H<1/2$ based on the well-posedness of the problem. We make use of a sequence of approximate solutions with the fractional Brownian noise replaced by its piecewise constant discretization to construct the finite element approximations of the SPDEs. The error estimate of the approximations is derived through rigorous convergence analysis.

Motivation & Objective

• To develop a stable and convergent finite element method for semilinear elliptic SPDEs driven by fractional Brownian motion with Hurst parameter H < 1/2.
• To address the challenge of irregularity in the noise process due to H < 1/2, which invalidates standard stochastic calculus approaches.
• To construct a sequence of approximate solutions by replacing the fractional Brownian motion with its piecewise constant discretization.
• To establish rigorous error estimates for the finite element approximations through convergence analysis.
• To ensure the well-posedness of the underlying SPDE problem as a foundation for numerical approximation.

Proposed method

• The method replaces the fractional Brownian motion in the SPDE with a piecewise constant approximation over time partitions to enable numerical treatment.
• Finite element spaces are constructed on a spatial mesh to discretize the elliptic operator in the SPDE.
• The resulting discrete problem is solved using Galerkin's method, leading to a system of random algebraic equations.
• Convergence analysis is performed in a suitable probabilistic and Sobolev-type norm to derive error bounds.
• The analysis relies on the well-posedness of the original SPDE, ensuring existence and uniqueness of the solution.
• Error estimates are derived by comparing the finite element solution to the exact solution, accounting for both spatial and noise discretization errors.

Experimental results

Research questions

• RQ1 How can finite element methods be adapted to solve semilinear elliptic SPDEs driven by fractional Brownian motion with H < 1/2?
• RQ2 What is the convergence behavior of finite element approximations when the noise is irregular due to H < 1/2?
• RQ3 Can a piecewise constant approximation of the fractional noise yield a stable and accurate numerical scheme for such SPDEs?
• RQ4 What error bounds can be rigorously established for the finite element solution in this non-Markovian, non-semimartingale setting?
• RQ5 How does the choice of spatial and temporal discretization affect the convergence rate of the numerical solution?

Key findings

• The finite element method with piecewise constant noise approximation yields a convergent scheme for semilinear elliptic SPDEs with H < 1/2.
• The error estimate for the finite element approximation is derived through rigorous convergence analysis in a suitable probabilistic norm.
• The method is valid under the assumption of well-posedness of the original SPDE, which is assumed to hold.
• The convergence analysis accounts for both spatial discretization and the approximation of the fractional noise.
• The error bound depends on the mesh size and the time step used in the piecewise constant approximation of the fractional Brownian motion.
• The approach provides a viable numerical framework for SPDEs driven by rough fractional Brownian motion where standard Itô calculus does not apply.