On weak solutions to stochastic differential inclusions driven by semimartingales

Stochastic Analysis and Applications

Volumn 22, Issue 5, 2004, Pages 1341-1361

  Michta, Mariusz ^a,b  

^a UNIVERSITY OF ZIELONA GÓRA   (Poland)

^b Inst Phys Fac Math Phys T   (Poland)

Author keywords

Martingale problem; Semimartingale; Stochastic differential inclusions; Weak convergence of probability measures; Weak solutions

Indexed keywords

EID: 5644257295     PISSN: 07362994     EISSN: None     Source Type: Journal    
DOI: 10.1081/SAP-200026471     Document Type: Article

References (28)

1. Ahmed, N.U. Nonlinear stochastic differential inclusions on Banach space. Stochast Anal. Appl. 1994, 12 (1), 1-10.
2. 0 semigroups and stochastic evolution inclusions. Discrete Math. DICO 2002, 22 (1), 125-149.
3. Ahmed, N.U. Optimal relaxed controls for nonlinear infinite dimensional stochastic differential inclusions. Opt. Control Differential Equations, M. Dekker Lect. Notes 1994, 160, 1-19.
4. Ahmed, N.U. Optimal relaxed controls for infinite dimensional stochastic systems of Zakai type. SIAM J. Contr. Optim. 1996, 34 (5), 1592-1615.
5. Aubin, J.P.; Frankowska, H. Set-Valued Analysis; Birkhauser: Boston, 1990.
6. Aubin, J.P.; Da Prato, G. Stochastic Nagumo's viability theorem. Stochast Anal. Appl. 1995, 13 (1), 1-11.
7. Bichteler, K. Stochastic Integration with Jumps; Cambridge Univ. Press, 2002.
8. Billingsley, P. Convergence of Probability Measures; Wiley: New York, 1968.
9. Fryszkowski, A. Continuous selections for a class of nonconvex multivalued maps. Stud. Math. 1983, 76, 163-174.
10. Hiai, F.; Umegaki, H. Integrals, conditional expectations, and martingales of multivalued functions. J. Multivar. Anal. 1977, 7, 149-182.
11. Hu, S.; Papageorgiou, N. Handbook of Multivalued Analysis, Theory; Kluwer: Boston, 1997; Vol. 1.
12. Jacod, J. Weak and strong solutions of stochastic differential equations. Stochastics 1980, 3, 171-191.
13. Jacod, J.; Memin, J. Weak and strong solutions of stochastic diffrential equations: Existence and stability. Lect. Notes in Math. 1981, 851, 169-212.
14. Jacod, J.; Shiryaev, A.N. Limit Theorems for Stochastic Processes; Springer: New York, 1987.
15. Kisielewicz, M.; Michta, M.; Motyl, J. Set-valued approach to stochastic cocontrol. Parts I and II. Dynamic. Syst. Appl. 2003, 12 (3-4), 405-467.
16. Kisielewicz, M. Differential Inclusions and Optimal Control; Kluwer Acad. Publ. and Polish Sci. Publ.: Warszawa-Dordrecht, 1991.
17. Kisielewicz, M. Set-valued stochastic integrals and stochastic inclusions. Stochast Anal. Appl. 1997, 15 (5), 783-800.
18. Kisielewicz, M. Weak compacteness of solution sets to stochastic differential inclusions with convex right-hand side. Topol. Meth. Nonlin. Anal. 2001, 18, 149-169.
19. Kurtz, T.G.; Protter, P. Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 1991, 19 (3), 1035-1070.
20. Levakov, A.A. Stochastic differential inclusions. J. Differential Equation 1997, 2 (33), 212-221.
21. Michta, M.; Motyl, J. Second order stochastic inclusions. Stochast Anal. Appl. 2004, 22 (3), 721-738.
22. Michta, M. Stochastic inclusions with multivalued integrators. Stochast Anal. Appl. 2002, 20 (4), 847-862.
23. Michta, M. Optimal solutions to stochastic differential inclusions. App. Math. 2002, 29 (4), 387-398.
24. Motyl, J. Stochastic functional inclusion driven by semimartingale. Stochast Anal. Appl. 1998, 16 (3), 517-532.
25. Motyl, J. Existence of solutions of set-valued Itô equation. Bull. Acad. Pol. Sci. 1998, 46, 419-430.
26. Protter, P. Stochastic Integration and Differential Equations: A New Approach; Springer: New York, 1990.
27. Słomiński, L. Stability of stochastic differential equations driven by general semimartingales. Dissertationes Math. 1996, 349, 1-109.
28. Stroock, D.; Varadhan, S.R. Multidimensional Diffusion Processes; Springer, 1975.