On the stochastic Magnus expansion and its application to SPDEs

Understand

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 It\^o sense, with progressively measurable coefficients, for which an explicit It\^o-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 t.
• 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.