Abstract
We derive the stochastic version of the Magnus expansion for linear systems of stochastic differential equations (SDEs). The main novelty with respect to the related literature is that we consider SDEs in the Itô sense, with progressively measurable coefficients, for which an explicit Itô-Stratonovich conversion is not available. We prove convergence of the Magnus expansion up to a stopping time tau and provide a novel asymptotic estimate of the cumulative distribution function of tau . As an application, we propose a new method for the numerical solution of stochastic partial differential equations (SPDEs) based on spatial discretization and application of the stochastic Magnus expansion. A notable feature of the method is that it is fully parallelizable. We also present numerical tests in order to asses the accuracy of the numerical schemes.
Highlights
The Magnus expansion is a classical tool to solve nonautonomous linear differential equations
We explore possible applications to the numerical solution of stochastic partial differential equations (SPDEs)
MEs as novel approximation tools for SPDEs; we study the error of this approximating procedure only numerically, in a case where an explicit benchmark is available, and we defer the theoretical error analysis to further studies
Summary
The Magnus expansion (hereafter referred to as ME) is a classical tool to solve nonautonomous linear differential equations.
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