Abstract
На плоском n-мерном торе изучаются стохастические дифференциальные включения с производными в среднем, у которых правые части имеют, вообще говоря, не выпуклые (асферические) значения. Выделен подкласс таких включений, для которых существует последовательность $\varepsilon$-аппроксимаций, поточечно сходящаяся к измеримому по Борелю селектору. На этой основе получена теорема существования решения.
Highlights
The concept of mean derivatives was introduced by E
Nelson in the 60-s years of 20th century for the needs of the so-called stochastic mechanics constructed by him
It turned out that equations and inclusions with mean derivatives naturally arose in many branches of mathematical physics, economics and other sciences
Summary
The concept of mean derivatives was introduced by E. Differential inclusions with mean derivatives, for which the right-hand sides are, generally speaking, have non-convex values of points These are mappings that are aspherical in all dimensions from 1 to n − 1 (see exact definitions below). For a subclass of mappings with this property on the flat n-dimensional torus an existence of solution theorem is proved for differential inclusions with mean derivatives It should be pointed out, that no unform but only point-wise convergence of ε-approximations of the right-hand sides of the ordinary differential inclusion, gives nothing useful for the investigation of those inclusions. This paper shows that in the case of stochastic differential inclusions the point-wise convergence of ε-approximations of the right-hand sides is a powerful machinery for proving the existence of solution theorems.
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