# Weak solutions of stochastic differential inclusions and their compactness

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2009)

- Volume: 29, Issue: 1, page 91-106
- ISSN: 1509-9407

## Access Full Article

top## Abstract

top## How to cite

topMariusz Michta. "Weak solutions of stochastic differential inclusions and their compactness." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 29.1 (2009): 91-106. <http://eudml.org/doc/271187>.

@article{MariuszMichta2009,

abstract = {In this paper, we consider weak solutions to stochastic inclusions driven by a semimartingale and a martingale problem formulated for such inclusions. Using this we analyze compactness of the set of solutions. The paper extends some earlier results known for stochastic differential inclusions driven by a diffusion process.},

author = {Mariusz Michta},

journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},

keywords = {semimartingale; stochastic differential inclusions; weak solutions; martingale problem; weak convergence of probability measures},

language = {eng},

number = {1},

pages = {91-106},

title = {Weak solutions of stochastic differential inclusions and their compactness},

url = {http://eudml.org/doc/271187},

volume = {29},

year = {2009},

}

TY - JOUR

AU - Mariusz Michta

TI - Weak solutions of stochastic differential inclusions and their compactness

JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization

PY - 2009

VL - 29

IS - 1

SP - 91

EP - 106

AB - In this paper, we consider weak solutions to stochastic inclusions driven by a semimartingale and a martingale problem formulated for such inclusions. Using this we analyze compactness of the set of solutions. The paper extends some earlier results known for stochastic differential inclusions driven by a diffusion process.

LA - eng

KW - semimartingale; stochastic differential inclusions; weak solutions; martingale problem; weak convergence of probability measures

UR - http://eudml.org/doc/271187

ER -

## References

top- [1] N.U. Ahmed, Nonlinear stochastic differential inclusions on Banach space, Stoch. Anal. Appl. 12 (1) (1994), 1-10. Zbl0789.60052
- [2] N.U. Ahmed, Impulsive perturbation of C₀ semigroups and stochastic evolution inclusions, Discuss. Math. DICO 22 (1) (2002), 125-149. Zbl1039.34055
- [3] N.U. Ahmed, Optimal relaxed controls for nonlinear infinite dimensional stochastic differential inclusions, Optimal Control of Differential Equations, M. Dekker Lect. Notes. 160 (1994), 1-19. Zbl0854.49006
- [4] N.U. Ahmed, Optimal relaxed controls for infinite dimensional stochastic systems of Zakai type, SIAM J. Contr. Optim. 34 (5) (1996), 1592-1615. Zbl0861.93030
- [5] P. Billingsley, Convergence of Probability Measures, Wiley, New York, 1968. Zbl0172.21201
- [6] S. Hu and N. Papageorgiou, Handbook of Multivalued Analysis, Vol. 1, Theory, Kluwer, Boston, 1997. Zbl0887.47001
- [7] J. Jacod, Weak and strong solutions of stochastic differential equations, Stochastics 3 (1980), 171-191. Zbl0434.60061
- [8] J. Jacod, A.N. Shiryaev, Limit Theorems for Stochastic Processes. Springer, New York, 1987. Zbl0635.60021
- [9] M. Kisielewicz, M. Michta, J. Motyl, Set-valued approach to stochastic control. Parts I, II, Dynamic. Syst. Appl. 12 (3&4) (2003), 405-466. Zbl1064.93042
- [10] M. Kisielewicz, Quasi-retractive representation of solution set to stochastic inclusions, J. Appl. Math. Stochastic Anal. 10 (3) (1997), 227-238. Zbl1043.34505
- [11] M. Kisielewicz, Set-valued stochastic integrals and stochastic inclusions, Stoch. Anal. Appl. 15 (5) (1997), 783-800. Zbl0891.93070
- [12] M. Kisielewicz, Stochastic differential inclusions, Discuss. Math. Differential Incl. 17 (1-2) (1997), 51-65. Zbl0911.93049
- [13] M. Kisielewicz, Weak compactness of solution sets to stochastic differential inclusions with convex right-hand side, Topol. Meth. Nonlin. Anal. 18 (2003), 149-169. Zbl1139.60331
- [14] M. Kisielewicz, Weak compactness of solution sets to stochastic differential inclusions with non-convex right-hand sides, Stoch. Anal. Appl. 23 (5) (2005), 871-901. Zbl1139.60332
- [15] M. Kisielewicz, Stochastic differential inclusions and diffusion processes, J. Math. Anal. Appl. 334 (2) (2007), 1039-1054. Zbl1123.60059
- [16] A.A. Levakov, Stochastic differential inclusions, J. Differ. Eq. 2 (33) (2003), 212-221. Zbl0911.60053
- [17] M. Michta, On weak solutions to stochastic differential inclusions driven by semimartingales, Stoch. Anal. Appl. 22 (5) (2004), 1341-1361. Zbl1059.93125
- [18] M. Michta, Optimal solutions to stochastic differential inclusions, Applicationes Math. 29 (4) (2002), 387-398. Zbl1044.93062
- [19] M. Michta and J. Motyl, High order stochastic inclusions and their applications, Stoch. Anal. Appl. 23 (2005), 401-420. Zbl1074.93034
- [20] J. Motyl, Stochastic functional inclusion driven by semimartingale, Stoch. Anal. Appl. 16 (3) (1998), 517-532. Zbl0914.60024
- [21] J. Motyl, Existence of solutions of set-valued Itô equation, Bull. Acad. Pol. Sci. 46 (1998), 419-430. Zbl0916.93069
- [22] P. Protter, Stochastic Integration and Differential Equations: A New Approach, Springer, New York, 1990.
- [23] L. Słomiński, Stability of stochastic differential equations driven by general semimartingales, Dissertationes Math. 349 (1996), 1-109.
- [24] C. Stricker, Loi de semimartingales et critéres de compacité, Sem. de Probab. XIX Lecture Notes in Math. 1123 (1985), Springer Berlin.
- [25] D. Stroock and S.R. Varadhan, Multidimensional Diffusion Processes, Springer, 1975. Zbl0426.60069