We consider stochastic approximation processes with constant step size whose associated deterministic system is an upper semicontinous differential inclusion. We prove that over any finite time span, the sample paths of the stochastic process are closely approximated by a solution of the differential inclusion with high probability. We then analyze infinite horizon behavior, showing that if the...

We consider a class of deterministic and stochastic dynamical systems with discontinuous drift f and solutions that are constrained to live in a given closed domain G in Rn according to a constraint vector field D(·) specified on the boundary ∂G of the domain. Specifically, we consider equations of the form φ = θ + η + u, θ̇ (t) ∈ F(φ(t)), a.e. t for u in an appropriate class of functions, where...

In this paper, we shall firstly illustrate why we should discuss the Aumann type set-valued Lebesgue integral of a set-valued stochastic process with respect to time t under the condition that the set-valued stochastic process takes nonempty compact subset of d-dimensional Euclidean space. After recalling some basic results about set-valued stochastic processes, we shall secondly prove that the...

The existence of Hilfer fractional stochastic Volterra–Fredholm integro-differential inclusions via almost sectorial operators is the topic our paper. researchers used calculus, analysis theory, and Bohnenblust–Karlin’s fixed point theorem for multivalued maps to support their findings. To begin with, we must establish a mild solution. In addition, show principle, an application presented.

We present a new Aumann-like integral for Hölder multifunction with respect to signal, based on the Young of particular set selections. This restricted Aumann has continuity properties that allow numerical approximation as well an existence theorem abstract stochastic differential inclusion. is applied concrete examples first order and second inclusions directed by fractional Brownian motion.

We study the existence of mild solutions for a class of impulsive fractional partial neutral stochastic integro-differential inclusions with statedependent delay. We assume that the undelayed part generates a solution operator and transform it into an integral equation. Sufficient conditions for the existence of solutions are derived by using the nonlinear alternative of Leray-Schauder type for...

it is known that a stochastic dierential equation (sde) induces two probabilisticobjects, namely a diusion process and a stochastic ow. while the diusion process isdetermined by the innitesimal mean and variance given by the coecients of the sde,this is not the case for the stochastic ow induced by the sde. in order to characterize thestochastic ow uniquely the innitesimal covariance give...