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On the stochastic Magnus expansion and its application to SPDEs
We derive a stochastic version of the Magnus expansion for the solution of
linear systems of It\^o stochastic differential equations (SDEs). The goal of
this paper is twofold. First, we prove existence and a representation formula
for the logarithm associated to the solution of the matrix-valued SDEs. Second,
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