Decomposition of stochastic flows generated by Stratonovich SDEs with jumps

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

Decomposition of stochastic flows generated by Stratonovich SDEs with jumps

Alison M. Melo ^1, , Leandro B. Morgado ^1, and Paulo R. Ruffino ^1,

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

Received October 2015 Revised March 2016 Published October 2016

Abstract
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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.

Keywords: decomposition of flows, group of diffeomorphisms, Marcus equation., Semimartingales with jumps, stochastic flows, smooth distributions.

Mathematics Subject Classification: Primary: 60H10, 58J65; Secondary: 58D0.

Citation: Alison M. Melo, Leandro B. Morgado, Paulo R. Ruffino. Decomposition of stochastic flows generated by Stratonovich SDEs with jumps. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3209-3218. doi: 10.3934/dcdsb.2016094

References:

[1]	D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge University Press, 2004. doi: 10.1017/CBO9780511755323.
[2]	J. M. Bismut, Mécanique Aléatoire, Lecture Notes in Math., 866. Springer-Verlag, Berlin-New York, 1981.
[3]	P. J. Catuogno, F. B. da Silva and P. R. Ruffino, Decomposition of stochastic flows with automorphism of subbundles component, Stochastics and Dynamics, 12 (2012), 1150013, 15 pp. doi: 10.1142/S0219493712003705.
[4]	P. J. Catuogno, F. B. da Silva and P. R. Ruffino, Decomposition of stochastic flows in manifolds with complementary distributions, Stochastics and Dynamics, 13 (2013), 1350009, 12 pp. doi: 10.1142/S0219493713500093.
[5]	F. Colonius and W. Kliemann, The Dynamics of Control, Systems and Control: Foundations and Applications. Birkhäuser Boston, Inc., Boston, MA, 2000. doi: 10.1007/978-1-4612-1350-5.
[6]	F. Colonius and P. R. Ruffino, Nonlinear Iwasawa decomposition of control flows, Discrete Continuous Dyn. Systems, (Serie A) 18 (2007), 339-354. doi: 10.3934/dcds.2007.18.339.
[7]	I. G. Gargate and P. R. Ruffino, An averaging principle for diffusions in foliated spaces, The Annals of Probab., 44 (2016), 567-588. doi: 10.1214/14-AOP982.
[8]	H. Kunita, Stochastic differential equations and stochastic flows of diffeomorphisms, in: École d'Eté de Probabilités de Saint-Flour XII - 1982, ed. P.L. Hennequin, Lecture Notes on Maths., Springer, 1097 (1984), 143-303. doi: 10.1007/BFb0099433.
[9]	H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, Cambridge, 1997.
[10]	X.-M. Li, An averaging principle for a completely integrable stochastic Hamiltonian system, Nonlinearity, 21 (2008), 803-822. doi: 10.1088/0951-7715/21/4/008.
[11]	T. Kurtz, E. Pardoux and P. Protter, Stratonovich stochastic differential equations driven by general semimartingales, Annales de l'I.H.P., section B, 31 (1995), 351-377.
[12]	M. Liao, Decomposition of stochastic flows and Lyapunov exponents, Probab. Theory Rel. Fields, 117 (2000), 589-607. doi: 10.1007/PL00008736.
[13]	J. Milnor, Remarks on infinite dimensional lie groups, In: Relativity, Groups and Topology II, Les Houches Session XL, 1983, North Holland, Amsterdam, 1984, 1007-1057.
[14]	K.-H. Neeb, Infinite Dimesional Lie Groups, Monastir Summer Schoool, 2009. Available from hal.archives-ouvertes.fr/docs/00/39/.../CoursKarl-HermannNeeb.pdf.
[15]	H. Omori, Infinite Dimensional Lie Groups, Translations of Math Monographs 158, AMS, 1997.
[16]	L. Morgado and P. Ruffino, Extension of time for decomposition of stochastic flows in spaces with complementary foliations, Electron. Comm. Probab., 20 (2015), 9pp. doi: 10.1214/ECP.v20-3762.
[17]	P. Ruffino, Decomposition of stochastic flow and rotation matrix, Stochastics and Dynamics, 2 (2002), 93-108. doi: 10.1142/S0219493702000327.
[18]	M. Patrão and L. A. B. San Martin, Semiflows on topological spaces: Chain transitivity and semigroups, J. Dynam. Differential Equations 19 (2007), 155-180. doi: 10.1007/s10884-006-9032-3.
[19]	K. Twardowska, Approximation theorems of Wong-Zakai type for stochastic differential equations in infinite dimensions, Dissertationes Math., (Rozprawy Mat.) 325 (1993), 54 pp.

show all references

References:

[1]	D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge University Press, 2004. doi: 10.1017/CBO9780511755323.
[2]	J. M. Bismut, Mécanique Aléatoire, Lecture Notes in Math., 866. Springer-Verlag, Berlin-New York, 1981.
[3]	P. J. Catuogno, F. B. da Silva and P. R. Ruffino, Decomposition of stochastic flows with automorphism of subbundles component, Stochastics and Dynamics, 12 (2012), 1150013, 15 pp. doi: 10.1142/S0219493712003705.
[4]	P. J. Catuogno, F. B. da Silva and P. R. Ruffino, Decomposition of stochastic flows in manifolds with complementary distributions, Stochastics and Dynamics, 13 (2013), 1350009, 12 pp. doi: 10.1142/S0219493713500093.
[5]	F. Colonius and W. Kliemann, The Dynamics of Control, Systems and Control: Foundations and Applications. Birkhäuser Boston, Inc., Boston, MA, 2000. doi: 10.1007/978-1-4612-1350-5.
[6]	F. Colonius and P. R. Ruffino, Nonlinear Iwasawa decomposition of control flows, Discrete Continuous Dyn. Systems, (Serie A) 18 (2007), 339-354. doi: 10.3934/dcds.2007.18.339.
[7]	I. G. Gargate and P. R. Ruffino, An averaging principle for diffusions in foliated spaces, The Annals of Probab., 44 (2016), 567-588. doi: 10.1214/14-AOP982.
[8]	H. Kunita, Stochastic differential equations and stochastic flows of diffeomorphisms, in: École d'Eté de Probabilités de Saint-Flour XII - 1982, ed. P.L. Hennequin, Lecture Notes on Maths., Springer, 1097 (1984), 143-303. doi: 10.1007/BFb0099433.
[9]	H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, Cambridge, 1997.
[10]	X.-M. Li, An averaging principle for a completely integrable stochastic Hamiltonian system, Nonlinearity, 21 (2008), 803-822. doi: 10.1088/0951-7715/21/4/008.
[11]	T. Kurtz, E. Pardoux and P. Protter, Stratonovich stochastic differential equations driven by general semimartingales, Annales de l'I.H.P., section B, 31 (1995), 351-377.
[12]	M. Liao, Decomposition of stochastic flows and Lyapunov exponents, Probab. Theory Rel. Fields, 117 (2000), 589-607. doi: 10.1007/PL00008736.
[13]	J. Milnor, Remarks on infinite dimensional lie groups, In: Relativity, Groups and Topology II, Les Houches Session XL, 1983, North Holland, Amsterdam, 1984, 1007-1057.
[14]	K.-H. Neeb, Infinite Dimesional Lie Groups, Monastir Summer Schoool, 2009. Available from hal.archives-ouvertes.fr/docs/00/39/.../CoursKarl-HermannNeeb.pdf.
[15]	H. Omori, Infinite Dimensional Lie Groups, Translations of Math Monographs 158, AMS, 1997.
[16]	L. Morgado and P. Ruffino, Extension of time for decomposition of stochastic flows in spaces with complementary foliations, Electron. Comm. Probab., 20 (2015), 9pp. doi: 10.1214/ECP.v20-3762.
[17]	P. Ruffino, Decomposition of stochastic flow and rotation matrix, Stochastics and Dynamics, 2 (2002), 93-108. doi: 10.1142/S0219493702000327.
[18]	M. Patrão and L. A. B. San Martin, Semiflows on topological spaces: Chain transitivity and semigroups, J. Dynam. Differential Equations 19 (2007), 155-180. doi: 10.1007/s10884-006-9032-3.
[19]	K. Twardowska, Approximation theorems of Wong-Zakai type for stochastic differential equations in infinite dimensions, Dissertationes Math., (Rozprawy Mat.) 325 (1993), 54 pp.

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