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

Minoo Kamrani, Kristian Debrabant, Nahid Jamshidi

TL;DR

This work addresses stiff stochastic differential equations driven by additive fractional Brownian noise with $H>\tfrac{1}{2}$, a setting arising from spatial discretizations of SPDEs. It extends the exponential Euler method to these fractional SDEs and proves strong convergence with a rate near the Hurst parameter $H$, independently of stiffness, by employing a variation-of-constants formulation and stochastic integral bounds. The paper also analyzes pathwise stability, showing that the discretization inherits a unique stationary solution that is pathwise attracting for sufficiently small step sizes, leveraging a fractional Ornstein-Uhlenbeck framework. Numerical experiments corroborate the theoretical findings, revealing near-$1$ convergence order and superior stability compared to standard Euler methods in stiff regimes.

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>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, M.G. Riedler, and P.E. Kloeden 2011), we conclude the existence of a unique stationary solution of the exponential Euler scheme that is pathwise asymptotically stable.

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

TL;DR

This work addresses stiff stochastic differential equations driven by additive fractional Brownian noise with , a setting arising from spatial discretizations of SPDEs. It extends the exponential Euler method to these fractional SDEs and proves strong convergence with a rate near the Hurst parameter , independently of stiffness, by employing a variation-of-constants formulation and stochastic integral bounds. The paper also analyzes pathwise stability, showing that the discretization inherits a unique stationary solution that is pathwise attracting for sufficiently small step sizes, leveraging a fractional Ornstein-Uhlenbeck framework. Numerical experiments corroborate the theoretical findings, revealing near- convergence order and superior stability compared to standard Euler methods in stiff regimes.

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>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, M.G. Riedler, and P.E. Kloeden 2011), we conclude the existence of a unique stationary solution of the exponential Euler scheme that is pathwise asymptotically stable.
Paper Structure (7 sections, 7 theorems, 57 equations, 2 figures)

This paper contains 7 sections, 7 theorems, 57 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Setting of the problem
  3. Convergence rate of the exponential Euler method
  4. Pathwise stability
  5. Fractional Ornstein-Uhlenbeck process
  6. Pathwise stability for exponential Euler method
  7. Numerical experiment

Key Result

Lemma 2.1

MR1741154 If $f,g \in L^2_{\phi}(\mathbb R^+),$ then $\int_0^{\infty} f(u) {\mathop{\mathrm{d}}\nolimits}B^H(u)$ and $\int_0^{\infty} g(u) {\mathop{\mathrm{d}}\nolimits}B^H(u)$ are well defined zero mean, Gaussian random variables such that

Figures (2)

  • Figure 1: Mean square errors of the exponential Euler method for different Hurst parameters $H$. $p$ denotes the respective numerically approximated convergence order.
  • Figure 2: Mean of 1000 simulated paths of the standard Euler method with $H=0.6$ and step length 1/256.

Theorems & Definitions (12)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.3
  • ...and 2 more