# American Institute of Mathematical Sciences

November  2016, 21(9): 3209-3218. doi: 10.3934/dcdsb.2016094

## Decomposition of stochastic flows generated by Stratonovich SDEs with jumps

 1 Department of Mathematics, University of Campinas - UNICAMP, Campinas, 13083-859, Brazil, Brazil, Brazil

Received  October 2015 Revised  March 2016 Published  October 2016

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 Itô-Ventzel-Kunita formula for stochastic flows with jumps generated by classical Marcus equation (as in Kurtz, Pardoux and Protter [11]). The results here correspond to an extension of Catuogno, da Silva and Ruffino [4], where this decomposition was studied for the continuous case.
Citation: Alison M. Melo, Leandro B. Morgado, Paulo R. Ruffino. Decomposition of stochastic flows generated by Stratonovich SDEs with jumps. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3209-3218. doi: 10.3934/dcdsb.2016094
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