### Abstract

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 language | English |
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Number of pages | 20 |

Journal | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |

Volume | 471 |

Issue number | 2184 |

DOIs | |

Publication status | Published - Dec 2015 |

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

## Fingerprint Dive into the research topics of 'The exponential Lie series for continuous semimartingales'. Together they form a unique fingerprint.

## Profiles

## Anke Wiese

- School of Mathematical & Computer Sciences - Associate Professor
- School of Mathematical & Computer Sciences, Actuarial Mathematics & Statistics - Associate Professor

Person: Academic (Research & Teaching)

## Cite this

Ebrahimi-Fard, K., Malham, S. J., Patras, F., & Wiese, A. (2015). The exponential Lie series for continuous semimartingales.

*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*,*471*(2184). https://doi.org/10.1098/rspa.2015.0429