Magnus Methods for Stochastic Delay-Differential Equations

TL;DR

This paper develops Magnus-based numerical schemes for stochastic delay-differential equations, demonstrating their convergence, stability, and efficiency advantages over traditional methods through theoretical proofs and numerical experiments.

Contribution

It introduces novel Magnus-based integrators for SDDEs and SPDDEs, combining stochastic Magnus methods with Taylor schemes, and compares their performance with classical methods.

Findings

Magnus-based schemes achieve proven convergence orders.

The schemes are numerically stable under fine discretization.

Compared to Euler--Maruyama, the Magnus methods show better stability and efficiency.

Abstract

This paper introduces Magnus-based methods for solving stochastic delay-differential equations (SDDEs). We construct Magnus--Euler--Maruyama (MEM) and Magnus--Milstein (MM) schemes by combining stochastic Magnus integrators with Taylor methods for SDDEs. These schemes are applied incrementally between multiples of the delay times. We present proofs of their convergence orders and demonstrate these rates through numerical examples and error graphs. Among the examples, we apply the MEM and MM schemes to both linear and nonlinear problems. We also apply the MEM scheme to a stochastic partial delay-differential equation (SPDDE), comparing its performance with the traditional Euler--Maruyama (EM) method. Under fine spatial discretization, the MEM scheme remains numerically stable while the EM method becomes unstable, yielding a significant computational advantage.