# Stochastic ItÃ´ Inclusion with Upper Separated Multifunctions

Jerzy Motyl^{*}

Let set-valued functions $F:I\times \Omega \times R^d \rightarrow \text{ClConv} R^d$ and $G:I\times \Omega \times R^d \rightarrow \text{ClConv} R^{d\times d_1}$ and an ${\cal F}_0 $-measurable $d$-dimensional random variable $\xi $ be given. Consider the following stochastic differential inclusion \[dX(t)\in F(t,X(t))dt+G(t,X(t))dW(t),\;X(0)=\xi.\] Such type stochastic inclusions were investigated in the early 90s of the last century among others by N.U. Ahmed, J.P. Aubin and G. Da Prato, H. Frankowska, M. Kisielewicz, R. Pettersson and by the author. Usually, the existence of strong solutions was proved under Lipschitz or monotonicity type assumptions on their set-valued coefficients.\newline In the talk we consider product measurable set-valued functions $F$ and $G$ being lsc and upper separated with respect to their last variables for each fixed $(t,\omega )$ and satisfying classical growth conditions. For such multifunctions we prove the existence of strong solutions to the stochastic inclusion mentioned above.

Mathematics Subject Classification: 60H20 47H04

Keywords: Stochastic inclusion; set-valued function; order convex selection; CarathÃ©odory condition

Minisymposion: Analysis and Control of Evolution Equations and Inclusions