ArticleOriginal scientific text
Title
Set-valued Stratonovich integral
Authors 1, 1
Affiliations
- Faculty of Mathematics, Computer Science and Econometrics University of Zielona Góra Szafrana 4a, 65-516 Zielona Góra, Poland
Abstract
The purpose of the paper is to introduce a set-valued Stratonovich integral driven by a one-dimensional Brownian motion. We discuss the existence of this integral and investigate its properties.
Keywords
set-valued function, Hukuhara differential, selection of a set-valued map, semimartingale, Stratonovich integral
Bibliography
- J.P. Aubin, Dynamic Economic Theory, A viability Approach, Springer, Verlag, Berlin 1997.
- J.P. Aubin and A. Cellina, Differential Inclusions, Noordhoff, Leyden 1984.
- J.P. Aubin and H. Frankowska, Set-valued analysis, Birkhäuser Boston-Basel-Berlin 1990.
- H.T. Banks and M.Q. Jacobs, A diffenertial calculus for set-valued function, J. Math. Anal. Appl. 29 (1970), 246-272.
- F. Hiai and H. Umegaki, Integrals, conditional expectations, and martingales of multivalued functions, J. Multivar. Anal. 7 (1977), 149-182.
- M. Hukuhara, Intégration des applications measurables dont a valeur est un compact convexe, Funkcialaj Ekvacioj 10 (1967), 205-223.
- M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer Acad. Publ.-PWN, Dordrecht-Boston-London Warszawa 1991.
- M. Kisielewicz, Set-valued stochastic integrals and stochastic inclusions, Stoch. Anal. Appl. 15 (5) (1997), 783-800.
- M. Kisielewicz, M. Michta and J. Motyl, Set-valued approach to stochastic control. Existence and regularity properties, Dynamic Syst. Appl. 12 (3-4) (2003), 405-432.
- M. Kisielewicz, M. Michta and J. Motyl, Set-valued approach to stochastic control. Viability and semimartingale issues, Dynamic Syst. Appl. 12 (3-4) (2003), 433-466.
- V. Lakshmikhantam, T. Gnana Bhaskar and D. Vasundhara, Theory of Set Differential Equations in Metric Space, (preprint) (2004).
- P. Protter, Stochastic Integration and Differential Equations: A New Approach, Springer, New York 1990.
- J. San Martin, One-dimensional Stratonovich differential equations, Ann. Probab. 21 (1) (1993), 509-553.
- E. Wong and M. Zakai, On the convergence of ordinary integrals to stochastic integrals, Ann. Math. Statist. 36 (1965), 1560-1564.