Flows and stochastic Taylor series in Ito calculus

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.

References in corpus (4)

• Loday-type Algebras and the Rota-Baxter Relation
• Mixable Shuffles, Quasi-shuffles and Hopf Algebras
• The pre-Lie structure of the time-ordered exponential
• Levy Processes and Quasi-Shuffle Algebras

Cited by in corpus (9)

• The exponential Lie series for continuous semimartingales
• The geometry of the space of branched Rough Paths
• Quasi-shuffle algebras and renormalisation of rough differential equations
• Tropical time series, iterated-sums signatures and quasisymmetric functions
• Algebraic Structures and Stochastic Differential Equations driven by Levy processes
• Quasi-geometric rough paths and rough change of variable formula
• Branched Itô formula and natural Itô-Stratonovich isomorphism
• Surjections and double posets
• A theory of pictures for quasi-posets