Finite element approximations for second order stochastic differential equation driven by fractional Brownian motion
This work addresses the numerical approximation of stochastic differential equations with rough noise, which is important for applications in physics and finance, but the results are incremental as they extend existing finite element methods to a specific class of fractional noise.
The paper develops finite element approximations for a second-order stochastic differential equation driven by fractional Brownian motion with Hurst index H ≤ 1/2, and provides error estimates through convergence analysis.
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.