Exponential Euler schemes for systems of fractional SDEs arising from engineering SPDEs: Convergence analysis

Abstract This paper addresses the numerical solution of systems of stochastic differential equations (SDEs) driven by additive fractional Brownian motion with Hurst parameter , arising from the spatial discretization of semilinear stochastic partial differential equations (SPDEs) that model complex engineering systems with memory effects. Such systems include nonlinear heat conduction in materials, elastic structures under stochastic loading, and anomalous diffusion in fluids or porous media. We study three exponential Euler schemes for the resulting stiff SDE systems. One method, previously analyzed in Kamrani et al.for stiff SDEs with stiffness concentrated in the system matrix, was shown to converge with order , independent of stiffness. Numerical experiments indicated a higher convergence rate close to 1, motivating a rigorous convergence analysis using Malliavin calculus. For this method, we establish order 1, which is optimal since it exceeds the Hölder regularity of fractional Brownian motion. For the remaining two methods, only a stiff convergence order of can be proven. Furthermore, we compare the three schemes in terms of ease of implementation and accuracy. The proposed methods are efficient and directly applicable to simulation‐based engineering problems with stochastic memory effects.