Decomposition of stochastic flows generated by Stratonovich SDEs 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.