Optimal Solutions for Inclusions of Geometric Brownian Motion Type with Mean Derivatives

Abstract

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.