Stochastic expansions and Hopf algebras

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9 Citations (Scopus)


We study solutions to nonlinear stochastic differential systems driven by a multidimensional Wiener process. A useful algorithm for strongly simulating such stochastic systems is the Castell-Gaines method, which is based on the exponential Lie series. When the diffusion vector fields commute, it has been proved that, at low orders, this method is more accurate in the mean-square error than corresponding stochastic Taylor methods. However, it has also been shown that when the diffusion vector fields do not commute, this is not true for strong order one methods. Here, we prove that when there is no drift, and the diffusion vector fields do not commute, the exponential Lie series is usurped by the sinh-log series. In other words, the mean-square error associated with a numerical method based on the sinh-log series is always smaller than the corresponding stochastic Taylor error, in fact to all orders. Our proof uses the underlying Hopf algebra structure of these series, and a two-alphabet associative algebra of shuffle and concatenation operations. We illustrate the benefits of the proposed series in numerical studies. © 2009 The Royal Society.

Original languageEnglish
Pages (from-to)3729-3749
Number of pages21
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2112
Publication statusPublished - 8 Dec 2009


  • Hopf algebras
  • Numerical approximation
  • Stochastic expansions


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