On stochastic differential inclusions with current velocities

Existence of solution theorems are obtained for stochastic differential inclusions given in terms of the so-called current velocities (symmetric mean derivatives, a direct analogs of ordinary velocity of deterministic systems) and quadratic mean derivatives (giving information on the diffusion coefficient) on the flat n-dimensional torus. Right-hand sides in both the current velocity part and the quadratic part are set-valued but satisfy the conditions, under which they have smooth selectors. Then we can reduce the inclusion to an equation with current velocities for which existence of solutions is known in the case of smooth right-hand side.