Optimal solutions for inclusions of geometric Brownian motion type with mean derivatives

The idea of mean derivatives of stochastic processes was suggested by E. Nelson in 60-th years of XX century. Unlike ordinary derivatives, the mean derivatives are well-posed for a very broad class of stochastic processes and equations with mean derivatives naturally arise in many mathematical models of physics (in particular, E. Nelson introduced the mean derivatives for the needs of Stochastic Mechanics, a version of quantum mechanics). Inclusions with mean derivatives is a natural generalization of those equations in the case of feedback control or in motion in complicated media. The paper is devoted to a brief introduction into the theory of equations and inclusions with mean derivatives and to investigation of a special type of such inclusions called inclusions of geometric Brownian motion type. The existence of optimal solutions maximizing a certain cost criterion, is proved.