On the solutions of set-valued stochastic differential equations in M-type 2 Banach spaces. (English) Zbl 1198.60027
Authors’ abstract: In a certain Banach space called an M-type 2 Banach space (including Hilbert spaces), we consider a set-valued stochastic differential equation with a set-valued drift term and a single valued diffusion term, under the Lipschitz continuity conditions, and we prove the existence and uniqueness of strong solutions which are continuous in the Hausdorff distance.
Reviewer: Toader Morozan (Bucureşti)
MSC:
| 60H25 | Random operators and equations (aspects of stochastic analysis) |
| 60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |
| 26E25 | Set-valued functions |
Keywords:
M-type 2 Banach space; integrals of set-valued stochastic processes; set-valued stochastic differential equationReferences:
| [1] | J. P. Aubin and H. Frankowska, Set-valued analysis, Birkhäuser, Boston, Basel, Berlin, 1990. · Zbl 0713.49021 |
| [2] | J. P. Aubin and G. Da Prato, The viability theorem for stochastic differential inclusions, Stochastic Anal. Appl. 16 (1998), 1–15. · Zbl 0931.60059 · doi:10.1080/07362999808809512 |
| [3] | Z. Brzeźniak and A. Carroll, Approximations of the Wong-Zakai type for stochastic differential equations in M-type 2 Banach spaces with applications to loop spaces, Séminaire de Probabilitiés, XXXVII: 251–289, Lecture Notes in Math. 1832, Springer-Verlag, Berlin, 2003. · Zbl 1040.60047 |
| [4] | C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Math. 580, Springer-Verlag, Berlin, 1977. · Zbl 0346.46038 |
| [5] | D. A. Charalambos and C. B. Kim, Infinite dimensional analysis, Springer-Verlag, Berlin, 1994. · Zbl 0839.46001 |
| [6] | F. Hiai and H. Umegaki, Integrals, conditional expectations and martingales of multivalued functions, J. Multivariate. Anal. 7 (1977), 149–182. · Zbl 0368.60006 · doi:10.1016/0047-259X(77)90037-9 |
| [7] | F. Hiai, Convergence of conditional expectations and strong laws of large numbers for multivalued random variables, Trans. Amer. Math. Soc. 291 (1985), 613–627. · Zbl 0583.60007 · doi:10.2307/2000101 |
| [8] | N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, North-Holland Math. Library, 24, North-Holland Publishing Co. Amsterdam-New York, Kodansha, Ltd., Tokyo, 1981. · Zbl 0495.60005 |
| [9] | D. A. Kandilakis and N. S. Papageorgiou, On the existence of solutions for random differential inclusions in a Banach space, J. Math. Anal. Appl. 126 (1987), 11–23. · Zbl 0629.60075 · doi:10.1016/0022-247X(87)90070-9 |
| [10] | M. Kisielewicz, M. Michta and J. Motyl, Set valued approach to stochastic control, part I, Existence and regularity properties, Dynam. Systems Appl. 12 (2003), 405–432. · Zbl 1063.93047 |
| [11] | M. Kisielewicz, M. Michta and J. Motyl, Set valued approach to stochastic control, part II, Viability and semimartingale issues, Dynam. Systems Appl. 12 (2003), 433–466. · Zbl 1064.93042 |
| [12] | M. Kisielewicz, Set-valued stochastic integrals and stochastic inclusions, Stochastic Anal. Appl. 15 (1997), 780–800. · Zbl 0891.93070 · doi:10.1080/07362999708809507 |
| [13] | M. Kisielewicz, Properties of solution set of stochastic inclusions, J. Appl. Math. Stochastic Anal. 6 (1993), 217–236. · Zbl 0796.93106 · doi:10.1155/S1048953393000188 |
| [14] | M. Kisielewicz, Existence theorem for nonconvex stochastic inclusions, J. Appl. Math. Stochastic Anal. 7 (1994), 151–159. · Zbl 0817.93065 · doi:10.1155/S104895339400016X |
| [15] | M. Kisielewicz, Quasi-retractive representation of solution sets to stochastic inclusions, J. Appl. Math. Stochastic Anal. 10 (1997), 227–238. · Zbl 1043.34505 · doi:10.1155/S1048953397000294 |
| [16] | H. Kunita, Stochastic differential equations and stochastic flows of diffeomorphisms, École d’été de Probabilités de Saint-Flour, XII–1982, 143–303, Lecture Notes in Math. 1097, Springer-Verlag, Berlin, 1984. · Zbl 0554.60066 |
| [17] | S. Li, Y. Ogura and V. Kreinovich, Limit theorems and applications of set-valued and fuzzy set-valued random variables, Kluwer Academic Publishers, Dordrecht, 2002. · Zbl 1348.60003 |
| [18] | R. E. Megginson, An introduction to Banach space theory, Springer, New York, 1998. · Zbl 0910.46008 · doi:10.1007/978-1-4612-0603-3 |
| [19] | M. Michta, Set-valued random differential equations in Banach space, Discuss. Math. Differential Incl. 15 (1995), 191–200. · Zbl 0878.34013 |
| [20] | J. Motyl, On the solution of stochastic differential inclusion, J. Math. Anal. Appl. 192 (1995), 117–132. · Zbl 0826.60053 · doi:10.1006/jmaa.1995.1163 |
| [21] | Y. Ogura, On stochastic differential equations with set coefficients and the Black-Scholes model, Proceedings of the Eighth International Conference on Intelligent Technologies, 300–304, Sydney, 2007. |
| [22] | B. Øksendal, Stochastic differential equations, Springer-Verlag, 1998. |
| [23] | G. Pisier, Martingales with values in uniformly convex spaces, Israel J. Math. 20 (1976), 326–350. · Zbl 0344.46030 · doi:10.1007/BF02760337 |
| [24] | J. J. Uhl, Jr., The range of a vector-valued measure, Proc. Amer. Math. Soc. 23 (1969), 158–163. JSTOR: · Zbl 0182.46903 · doi:10.2307/2037509 |
| [25] | J. Zhang, S. Li, I. Mitoma and Y. Okazaki, On set-valued stochastic integrals in an M-type 2 Banach space, J. Math. Anal. Appl. 350 (2009), 216–233. · Zbl 1155.60021 · doi:10.1016/j.jmaa.2008.09.017 |
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