Decomposition of stochastic flows generated by Stratonovich SDEs with jumps

TL;DR

This paper develops a decomposition of stochastic flows with jumps on manifolds into horizontal and vertical components, providing stochastic differential equations for each, extending previous continuous case results to jumps.

Contribution

It introduces a novel decomposition of stochastic flows with jumps into components within infinite-dimensional Lie groups, with corresponding SDEs, extending prior continuous case work.

Findings

Decomposition of stochastic flows with jumps into horizontal and vertical parts.

Derived Stratonovich SDEs for each component in Lie groups.

Extended Itô-Ventzel-Kunita formula to flows with jumps.

Abstract

Consider a manifold $M$ endowed locally with a pair of complementary distributions $\Delta^H \oplus \Delta^V=TM$ and let $\text{Diff}(\Delta^H, M)$ and $\text{Diff}(\Delta^V, M)$ be the corresponding Lie subgroups generated by vector fields in the corresponding distributions. We decompose a stochastic flow with jumps, up to a stopping time, as $\varphi_t = \xi_t \circ \psi_t$, where $\xi_t \in \text{Diff}(\Delta^H, M)$ and $\psi_t \in \text{Diff}(\Delta^V, M)$. Our main result provides Stratonovich stochastic differential equations with jumps for each of these two components in the corresponding infinite dimensional Lie groups. We present an extension of the It\^o-Ventzel-Kunita formula for stochastic flows with jumps generated by classical Marcus equation (as in Kurtz, Pardoux and Protter, Annales de L'I.H.P. section B, 1995, among others). The results here correspond to an extension of Catuogno, da Silva and Ruffino, Stoch. Dyn. 2013, where this decomposition was studied for the continuous case.