Stochastic set differential equations

Nonlinear Analysis, Theory, Methods and Applications

Volumn 72, Issue 3-4, 2010, Pages 1247-1256

  Malinowski, Marek T ^a   Michta, Mariusz ^a  

^a UNIVERSITY OF ZIELONA GÓRA   (Poland)

Author keywords

Set valued stochastic integral; Set valued stochastic process; Stochastic set differential equation

Indexed keywords

A-STABILITY; CONTINUOUS DEPENDENCE; EXISTENCE AND UNIQUENESS OF SOLUTION; INITIAL CONDITIONS; SET DIFFERENTIAL EQUATIONS; STOCHASTIC INTEGRAL; STOCHASTIC PROCESS; WIENER PROCESS;

DIFFERENTIAL EQUATIONS; RANDOM PROCESSES; STABILITY;

STOCHASTIC SYSTEMS;

EID: 71549153153     PISSN: 0362546X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.na.2009.08.015     Document Type: Article

References (32)

1. De Blasi F.S., and Iervolino F. Equazioni differenziali con soluzioni a valore compatto convesso. Boll. Unione Mat. Ital. 4 2 (1969) 194-501
2. Diamond P., and Kloeden P. Metric Spaces of Fuzzy Sets: Theory and Applications (1994), World Scientific, Singapore
3. Kisielewicz M. Differential Inclusions and Optimal Control (1991), Kluwer Acad. Publ., Dordrecht
4. Lakshmikantham V., Lela S., and Vatsala A. Interconnection between set and fuzzy differential equations. Nonlinear Anal. 54 (2003) 351-360
5. Lakshmikantham V., and Mohapatra R.N. Theory of Fuzzy Differential Equations and Inclusions (2003), Taylor and Francis Publishers, London
6. Lakshmikantham V., and Tolstonogov A.A. Existence and interrelation between set and fuzzy differential equations. Nonlinear Anal. 55 (2003) 255-268
7. Lakshmikantham V., Gnana Bhaskar T., and Vasundhara Devi J. Theory of Set Differential Equations in a Metric Space (2006), Cambridge Scientific Publ.
8. Tolstonogov A. Differential Inclusions in a Banach Space (2000), Kluwer Acad. Publ., Dordrecht
9. Artstein Z. A calculus for set-valued maps and set-valued evolution equations. Set-valued Anal. 3 (1995) 216-261
10. De Blasi F.S. Banach-Saks-Mazur and Kakutani-Ky Fan theorems in spaces of multifunctions and applications to set differential inclusions. Dynamic Systems Appl. 16 1 (2007) 73-89
11. Brandão Lopes Pinto A.I., de Blasi F.S., and Iervolino F. Uniqueness and existence theorems for differential equations with convex valued solutions. Boll. Unione Mat. Ital. 4 3 (1970) 1-12
12. Galanis G.N., Gnana Bhaskar T., Lakshmikantham V., and Palamides P.K. Set valued functions in Frechet spaces: Continuity, Hukuhara differentiability and applications to set differential equations. Nonlinear Anal. 61 (2005) 559-575
13. Kisielewicz M. Description of a class of differential equations with set-valued solutions. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 8 58 (1975) 158-162
14. Kisielewicz M. Method of averaging for differential equations with compact convex valued solutions. Rend. Mat. 6 9 (1976) 397-408
15. Kisielewicz M., Serafin B., and Sosulski W. Existence theorem for functional differential equations with compact convex valued solutions. Demonstratio Math. 9 (1976) 229-237
16. Michta M. Set valued random differential equations in Banach spaces. Discuss. Math. Differential Incl. 15 (1995) 124-200
17. Michta M. Continuity properties of solutions of multivalued equations with white noise perturbation. J. Appl. Math. Stoch. Anal. 10 (1997) 239-248
18. Plotnikov A.V., and Plotnikova I. Averaging of equations of controlled motion in a metric space. Cybernet. Systems Anal. 33 (1997) 601-606
19. Plotnikov V.A., and Rashkov P.I. Averaging in differentials equations with Hukuhara derivative and delay. International Conference on Differential and Functional Differential Equations (Moscow, 1999). Funct. Differ. Equ. 8 (2001) 371-381
20. Ahmed N.U. Nonlinear stochastic differential inclusions on Banach spaces. Stoch. Anal. Appl. 12 1 (1994) 1-10
21. Aubin J.P., and Da Prato G. The viability theorem for stochastic differential inclusions. Stoch. Anal. Appl. 16 (1998) 1-15
22. Kisielewicz M. Set-valued stochastic integrals and stochastic inclusions. Stoch. Anal. Appl. 15 5 (1997) 783-800
23. Kisielewicz M., Michta M., and Motyl J. Set-valued approach to stochastic control. Part: I and II. Dynamic Syst. Appl. 12 3-4 (2003) 405-466
24. Michta M., and Motyl J. Differentiable selections of multifunctions and their applications. Nonlinear Anal. 66 2 (2007) 536-545
25. Kim B.K., and Kim J.H. Stochastic integrals of set-valued processes and fuzzy processes. J. Math. Anal. Appl. 236 (1999) 480-502
26. Jung E.J., and Kim J.H. On set-valued Stochastic integrals. Stoch. Anal. Appl. 21 2 (2003) 401-418
27. Li Sh., and Ren A. Representation theorems, set-valued and fuzzy set-valued Ito integral. Fuzzy Sets and Systems 158 (2007) 949-962
28. Hu S., and Papageorgiou N. Handbook of Multivalued Analysis. Theory vol.1 (1997), Kluwer Acad. Publ., Boston
29. Hiai F., and Umegaki H. Integrals, conditional expectations and martingales for multivalued functions. J. Multivariate Anal. 7 (1977) 147-182
30. Protter Ph. Stochastic Integration and Differential Equations: A New Approach (1990), Springer-Verlag, New York
31. Chung K.L., and Williams R.J. Introduction to Stochastic Integration (1983), Birkhäuser, Boston
32. Aubin J.P., and Cellina A. Differential Inclusions (1984), Springer, New York