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### Session S17 - Stochastic Systems: Analysis, Numerics and Applications

Friday, July 16, 14:15 ~ 14:50 UTC-3

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

### Paulo Ruffino

#### University of Campinas, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakd7d3a86b7bafb5025f582ebed64444bb').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyd7d3a86b7bafb5025f582ebed64444bb = 'r&#117;ff&#105;n&#111;' + '&#64;'; addyd7d3a86b7bafb5025f582ebed64444bb = addyd7d3a86b7bafb5025f582ebed64444bb + '&#117;n&#105;c&#97;mp' + '&#46;' + 'br'; var addy_textd7d3a86b7bafb5025f582ebed64444bb = 'r&#117;ff&#105;n&#111;' + '&#64;' + '&#117;n&#105;c&#97;mp' + '&#46;' + 'br';document.getElementById('cloakd7d3a86b7bafb5025f582ebed64444bb').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyd7d3a86b7bafb5025f582ebed64444bb + '\'>'+addy_textd7d3a86b7bafb5025f582ebed64444bb+'<\/a>';

Let $M$ be a compact manifold equipped with a pair of complementary foliations, say horizontal and vertical. In Melo, Morgado and Ruffino (Disc Cont Dyn Syst- B 2016, 21(9)) it is proved that if a semimartingale $X_t$ has a finite number of jumps in compact intervals then, up to a stopping time $\tau$, a stochastic flow of local diffeomorphisms in $M$ driven by $X_t$ can be decomposed into a process in the Lie group of diffeomorphisms which preserve the horizontal foliation composed with a process in the Lie group of diffeomorphisms which preserve the vertical foliation. SDEs of these processes are shown. The stochastic flows with jumps are generated by the classical Marcus equation (as in Kurtz, Pardoux and Protter, Annal. I.H.P., section B, 31 (1995)). Here we enlarge the scope of this geometric decomposition and consider flows driven by arbitrary semimartingales with jumps. Our technique is based in an extension of the Itô-Ventzel-Kunita formula for stochastic flows with jumps. Geometrical and others topological obstructions for the decomposition are also considered, e.g. sufficient conditions for the existence of global decomposition for all $t\geq 0$.

Joint work with Lourival Lima (University of Campinas, Brazil).