On existence of solution in $\mathbb{R}^n$ of stochastic differential inclusions with current velocities in the presence of approximations with uniformly bounded first partial derivatives

Notion of mean derivatives was introduced by Edward Nelson for the needs of stochastic mechanics (a version of quantum mechanics). Nelson introduced forward and backward mean derivatives while only their half-sum, symmetric mean derivative called current velocity, is a direct analog of ordinary velocity for deterministic processes. Another mean derivative called quadratic, was introduced by Yuri E. Gliklikh and Svetlana V. Azarina. It gives information on the diffusion coefficient of the process and using Nelson's and quadratic mean derivatives together, one can in principle recover the process from its mean derivatives. Since the current velocities are natural analogs of ordinary velocities of deterministic processes, investigation of equations and especially inclusions with current velocities is very much important for applications since there are a lot of models of various physical, economical etc. processes based on such equations and inclusions. 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 $\mathbb{R}^n$. Right-hand sides in both the current velocity part and the quadratic part are set-valued but satisfy some natural conditions.