The upper and lower solutions method for stochastic inclusions with discontinuous multivalued mappings

Stochastic Analysis and Applications

Volumn 26, Issue 6, 2008, Pages 1181-1200

  Michta, Mariusz ^a  

^a UNIVERSITY OF ZIELONA GÓRA   (Poland)

Author keywords

Fixed points; Increasing and upper separated multifunction; Stochastic inclusion; Upper and lower solution

Indexed keywords

EID: 55249110014     PISSN: 07362994     EISSN: 15329356     Source Type: Journal    
DOI: 10.1080/07362990802405711     Document Type: Article

References (32)

1. Agarwal, R.P., Dhage, B.C., and O'Regan, D. 2003. The upper and lower solution method for differential inclusions via a lattice fixed point theorem. Dynamic Systems and Applications 12:1-7.
2. Ahmed, N.U. 1994. Nonlinear stochastic differential inclusions on Banach spaces. Stochastic Analysis and Applications 12(1):1-10.
3. Ahmed, N.U. 1994. Optimal relaxed controls for nonlinear infinite dimensional stochastic differential inclusions. Marcel Dekker Lecture Notes. Optimal Control of Differential Equations 160:1-19.
4. Ahmed, N.U. 2002. Optimal control of impulsive stochastic evolution inclusions. Discussions in Mathematics DICO 22(2):155-184.
5. Ahmed, N.U., and Xiang, X. 2003. Differential inclusions on Banach spaces and their optimal control. Nonlinear Functional Analysis and Applications 8(3):461-488.
6. Ahmed, N.U., and Xiang, X. 2003. Optimal relaxed controls for differential inclusions on Banach spaces. Stochastic Analysis and Applications 12(1):1-10.
7. Aubin, J.P., and Cellina, A. 1984. Differential Inclusions. Noordhoff, Leyden.
8. Aubin, J.P., and Da Prato, G. 1998. The viability theorem for stochastic differential inclusions. Stochastic Analysis and Applications 16:1-15.
9. Dhage, B.C. 2007. Monotone increasing multi-valued random operators and differential inclusions. Nonlinear Functional Analysis and Applications 12(3):399-419.
10. Ding, X., and Wu, R. 1998. A new proof for comparison theorems for stochastic differential inequalities with respect to semimartingales. Stochastic Processes and Applications 78:155-171.
11. Heikkila, S. 2007. Fixed point results for multifunctions in ordered topological spaces with applications to inclusion problems and game theory. Dynamic Systems and Applications 16:105-120.
12. Halidias, N., and Kloeden, P.E. 2006. A note on strong solutions of stochastic differential equations with a discontinuous drift coefficient. JAMSA doi: 10.1155/JAMSA/2006/73257.
13. Halidias, N., and Michta, M. 2008. The method of upper and lower solutions of stochastic differential equations and applications. Stochastic Analysis and Applications 16(1):16-28.
14. Hiai, F., and Umegaki, H. 1977. Integrals, conditional expectations, and martingales of multivalued functions. Journal of Multivariate Analysis 7:149-182.
15. Hu, Sh., and Papageorgiou, N.S. 1997. Handbook of Multivalued Analysis. Vol. I. Kluwer, Dordrecht.
16. Jacod, J., and Memin, J. 1980. Weak and strong solutions of stochastic differential equations: existence and stability. In Lecture Notes in Math. Vol. 851. Springer, 169-212.
17. Ladde, G.S., Lakshmikantham, V., and Pachpatte, B.G. 1982. The method of upper, lower solutions and Volterra integral equations. Journal of Integral Equations 4(4):353-381.
18. Lakshmikantham, V. 1984. Monotone technology for nonlinear problems. Journal of Mathematics and Physical Science 18(1):65-72.
19. Karatzas, I., and Shreve, S. 1988. Brownian Motion and Stochastic Calculus. Springer Verlag, New York.
20. Kannan, R., and Lakshmikantham, V. 1983. Existence of periodic solutions of semilinear parabolic equations and the method of upper and lower solutions. JMAA 97 (1):291-299.
21. Kisielewicz, M. 1997. Set-valued stochastic integrals and stochastic inclusions. Stochastic Analysis and Applications 15(5):783-800.
22. Kisielewicz, M., Michta, M., and Motyl, J. 2003. Set-valued approach to stochastic control. Existence and regularity properties. Dynamic Systems and Applications 12(3-4):405-432.
23. Kisielewicz, M., Michta, M., and Motyl, J. 2003. Set-valued approach to stochastic control. Viability and semimartingale issues. Dynamic Systems and Applications 12(3-4):433-466.
24. Michta, M. 2002. Stochastic inclusions with multivalued integrators. Stochastic Analysis and Applications 20(4):847-862.
25. Michta, M., and Motyl, J. 2006. Convex selections of multifunctions and their applications. Optimization 55(1-2):91-99.
26. Michta, M., and Motyl, J. 2007. Set valued Stratonovich integral and Stratonovich type stochastic inclusion. Dynamic Systems and Applications 16:141-154.
27. Motyl, J. 1994. On the solution of stochastic differential inclusion. Journal of Mathematics and Analytical Applications 192:117-132.
28. Phelps, R.R. 1989. Convex Functions, Monotone Operators and Differentiability. Springer Verlag, Berlin.
29. Protter, P. 1990. Stochastic Integration and Differential Equations: A New Approach. Springer Verlag, New York.
30. Rockafellar, R.T., and Wets, R.J.-B. 1998. Variational Analysis. Springer Verlag, Berlin.
31. Rong, S. 1985. On strong solutions, uniqueness, stability and comparison theorems for a stochastic system with Poisson jumps. In Lecture Notes in Contra Inference Science 75. Thoma, M. (ed.), Springer, Berlin, pp. 352-381.
32. Zeidler, E. 1985. Nonlinear Functional Analysis and Its Applications I: Fixed Points Theorems. Springer-Verlag, New York.