Efficient strong integrators for linear stochastic systems

Gabriel Lord, S. J A Malham, Anke Wiese

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17 Citations (Scopus)


We present numerical schemes for the strong solution of linear stochastic differential equations driven by an arbitrary number of Wiener processes. These schemes are based on the Neumann (stochastic Taylor) and Magnus expansions. First, we consider the case when the governing linear diffusion vector fields commute with each other, but not with the linear drift vector field. We prove that numerical methods based on the Magnus expansion are more accurate in the mean-square sense than corresponding stochastic Taylor integration schemes. Second, we derive the maximal rate of convergence for arbitrary multidimensional stochastic integrals approximated by their conditional expectations. Consequently, for general nonlinear stochastic differential equations with noncommuting vector fields, we deduce explicit formulae for the relation between error and computational costs for methods of arbitrary order. Third, we consider the consequences in two numerical studies, one of which is an application arising in stochastic linear-quadratic optimal control. © 2008 Society for Industrial and Applied Mathematics.

Original languageEnglish
Pages (from-to)2892-2919
Number of pages28
JournalSIAM Journal on Numerical Analysis
Issue number6
Publication statusPublished - 2008


  • Linear stochastic differential equations
  • Magnus expansion
  • Stochastic linear-quadratic control
  • Strong numerical methods


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