Stochastic inclusions with current velocities having decomposable right-hand sides

An existence of solution theorem is 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, lower semi-continuous but not necessarily have convex images. Instead we suppose that they are decomposable.