Exponential Euler method for stiff stochastic differential equations with additive fractional Brownian noise

Abstract

We discuss a system of stochastic differential equations with a stiff linear term and additive noise driven by fractional Brownian motions (fBms) with Hurst parameter $H>\frac{1}{2}$, which arise e.g. from spatial approximations of stochastic partial differential equations. For their numerical approximation, we present an exponential Euler scheme and show that it converges in the strong sense with an exact rate close to the Hurst parameter H. Further, based on E. Buckwar et al. [The numerical stability of stochastic ordinary differential equations with additive noise, Stoch. Dyn. 11 (2011), pp. 265–281], we conclude the existence of a unique stationary solution of the exponential Euler scheme that is pathwise asymptotically stable.

KEYWORDS:

• Stiff stochastic differential equations
• fractional Brownian motion
• exponential Euler scheme
• pathwise stability

AMS SUBJECT CLASSIFICATIONS:

• 60H35
• 65C30

Disclosure statement

No potential conflict of interest was reported by the author(s).