### Abstract

For general stochastic systems driven by continuous semimartingales an explicit formula for the logarithm of the Itô flow map is given. The computation relies on the lift to quasi-shuffle algebras of formulas involving products of Itô integrals of semimartingales. Whereas the Chen–Strichartz formula computing the logarithm of the Stratonovich flow map is classically expanded as a formal sum indexed by permutations, the analogous formula in Itô calculus is naturally indexed by surjections. This reflects the change of algebraic background involved in the transition between the two integration theories. Lastly, we extend our formula for the quasi-shuffle Chen–Strichartz series for the logarithm of the flow map to the non-commutative case. For linear matrix-valued SDEs driven by arbitrary semimartingales we obtain a similar formula.

Original language | English |
---|---|

Article number | 495202 |

Number of pages | 17 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 48 |

Issue number | 49 |

DOIs | |

Publication status | Published - 18 Nov 2015 |

## Fingerprint Dive into the research topics of 'Flows and stochastic Taylor series in Ito calculus'. 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

Wiese, A., Malham, S. J., Ebrahimi-Fard, K., & Patras, F. (2015). Flows and stochastic Taylor series in Ito calculus.

*Journal of Physics A: Mathematical and Theoretical*,*48*(49), [495202]. https://doi.org/10.1088/1751-8113/48/49/495202