Flows and stochastic Taylor series in Ito calculus

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

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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 languageEnglish
Article number495202
Number of pages17
JournalJournal of Physics A: Mathematical and Theoretical
Volume48
Issue number49
DOIs
Publication statusPublished - 18 Nov 2015

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