Stochastic Inclusions with Forward Mean Derivatives Having Decomposable Right-Hand Sides

Abstract

In this paper, we prove a theorem on the existence of solutions for stochastic differential inclusions given in terms of the forward mean derivatives and the quadratic mean derivatives. These derivatives present information on the drift and the diffusion coefficient, respectively. The forward mean derivatives were introduced by E. Nelson for the needs of the so-called stochastic mechanics (a version of quantum mechanics), while the quadratic mean derivatives were introduced by Yu.E. Gliklich and S.V. Azarina. In the case of both the forward mean derivatives and the quadratic mean derivatives, we assume that the righthand side is set-valued and lower semi-continuous, but not necessarily convex. Instead of this, we assume that the right-hand side is decomposable. Such inclusions naturally arise in many models of physical processes.