Flows and stochastic Taylor series in Ito calculus

TL;DR

This paper derives explicit formulas for the logarithm of the Ito flow map in stochastic calculus, using algebraic structures like quasi-shuffle algebras, highlighting differences from Stratonovich calculus.

Contribution

It provides a novel explicit formula for the Ito flow map logarithm, employing quasi-shuffle algebra techniques and revealing algebraic distinctions from Stratonovich calculus.

Findings

Explicit formula for Ito flow map logarithm

Use of quasi-shuffle algebra in stochastic calculus

Indexing by surjections instead of permutations

Abstract

For stochastic systems driven by continuous semimartingales an explicit formula for the logarithm of the Ito flow map is given. A similar formula is also obtained for solutions of linear matrix-valued SDEs driven by arbitrary semimartingales. The computation relies on the lift to quasi-shuffle algebras of formulas involving products of Ito 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 Ito calculus is naturally indexed by surjections. This reflects the change of algebraic background involved in the transition between the two integration theories.