DIFFERENTIAL INCLUSIONS WITH MEAN DERIVATIVES

We introduce and investigate a new sort of stochastic differential inclusions given in terms of mean derivatives of a stochastic process, introduced by E. Nelson for the needs of the so called stochastic mechanics. This class of stochastic inclusions is ideologically the closest one to ordinary differential inclusions. We consider three types of inclusions: with forward mean derivatives, with backward mean derivatives and with current velocities (symmetric mean derivatives). These types have different properties and physical meaning. Some existence of solutions results are proved.