The exponential Lie series for continuous semimartingales

Kurusch Ebrahimi-Fard, Simon John Malham, Frederic Patras, Anke Wiese

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13 Citations (Scopus)
55 Downloads (Pure)


We consider stochastic differential systems driven by continuous semimartingales and governed by non-commuting vector fields. We prove that the logarithm of the flowmap is an exponential Lie series. This relies on a natural change of basis to vector fields for the associated quadratic covariation processes, analogous to Stratonovich corrections. The flowmap can then be expanded as a series in compositional powers of vector fields and the logarithm of the flowmap can thus be expanded in the Lie algebra of vector fields. Further, we give a direct explicit proof of the corresponding Chen--Strichartz formula which provides an explicit formula for the Lie series coefficients. Such exponential Lie series are important in the development of strong Lie group integration schemes that ensure approximate solutions themselves lie in any homogeneous manifold on which the solution evolves.
Original languageEnglish
Number of pages20
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2184
Publication statusPublished - Dec 2015


  • Ito stochastic flows
  • quasi-shuffle product
  • exponential Lie series
  • Chen-Strichartz formula

ASJC Scopus subject areas

  • Mathematics(all)


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