Set-valued stochastic integrals and stochastic inclusions in a plane

Władysław Sosulski

ArticleOriginal scientific text

Title

Set-valued stochastic integrals and stochastic inclusions in a plane

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Affiliations

Institute of Mathematics, University of Zielona Góra, Podgórna 50, 65-246 Zielona Góra, Poland

Abstract

We present the concepts of set-valued stochastic integrals in a plane and prove the existence of a solution to stochastic integral inclusions of the form

z_{s, t} \in φ_{s, t} + \int_{0}^{s} \int_{0}^{t} F_{u, v} (z_{u, v}) d u d v + \int_{0}^{s} \int_{0}^{t} G_{u, v} (z_{u, v}) d w_{u, v}

Keywords

ENG

stochastic inclusions in the plane, set-valued random field, two-parameter stochastic process, weak compactness

R. Cairoli, Sur une équation differentielle stochastique, Compte Rendus Acad. Sc. Paris 274 (A) (1972), 1739-1742.
R. Cairoli and J.B. Walsh, Stochastic integrals in the plane, Acta Mathematica 134 (1975), 112-183.
F. Hiai and H. Umegaki, Integrals, conditional expectations, and martingales of multivalued functions, J. Multivar. Anal. 7 (1977), 149-162.
M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer Acad. Publ. - PWN, Dordrecht, Boston - London, Warszawa 1991.
M. Kisielewicz, Properties of solution set of stochastic inclusions, J. Appl. Math. Stoch. Anal. 6 (3) (1993), 217-236.
L. Ponomarenco, Stochastic integral with respect to the multiparameter Brownian motion and attached stochastic equations (in Russian), Teor. Veroiatn. i Mat. Stat. Kiev 7 (1972) 100-109.
W. Sosulski, Subtrajectory integrals of set-valued functions depending on parameters, Discuss. Math. 10 (1990), 99-121.
T.J. Tsarenco, On some scheme of the construction of stochastic integral for the radom field (in Russian), Kibernetika 1 (1972), 113-118.
C. Tudor, Stochastic integral equations in the plane, Preprint Series in Mathematics, INCREST 29 (1979) 507-538.

Title

Set-valued stochastic integrals and stochastic inclusions in a plane

Affiliations

Abstract

Keywords

Bibliography