Monotone operators in stochastic set-valued equations

Let F and G be set-valued functions. Under certain measurability conditions there exist set-valued stochastic integrals integral (t)(s) Fd tau, and integral (t)(s) GdW tau as Aumann's type integrals. Given such integrals we study a stochastic inclusion of the form: x(t) - x(s) is an element of integral (t)(s) F(x)(tau)d tau + integral (t)(s) G(x)(tau)dW(tau). We find sufficient conditions for the existence of strong solutions to the inclusion which differ both from Lipschitz and Pardoux "monotone" conditions. Secondary, the viability property for such a type inclusion will be discussed.