In the paper, we consider functional set-valued integral equations whose representation contains integrals occurring symmetrically on both sides of equation. On coefficients equation, impose certain conditions, more general than standard Lipschitz condition, which allow application Bihari–LaSalle inequality in proofs obtained theorems. this way, obtain a result about existence and uniqueness so...

We study the small time path behavior of double stochastic integrals of the form ∫ t 0 ( ∫ r 0 b(u)dW (u)) dW (r), where W is a d-dimensional Brownian motion and b an integrable progressively measurable stochastic process taking values in the set of d× dmatrices. We prove a law of the iterated logarithm that holds for all bounded progressively measurable b and give additional results under cont...

For dynamical systems defined by vector fields over a compact invariant set, we introduce a new class of approximated first integrals based on finite-time averages and satisfying an explicit first order partial differential equation. Referring to the PDE framework, we detect their viscosity robustness with respect to stochastic perturbations of the vector field. We formulate this approximating ...

We consider a stochastic control problem where the set of controls is not necessarily convex and the system is governed by a nonlinear backward stochastic differential equation. We establish necessary as well as sufficient conditions of optimality for two models. The first concerns the strict (classical) controls. The second is an extension of the first to relaxed controls, who are a measure va...

In this paper we introduce the interval-valued APHenstock-Stieltjes integral and investigate some properties of the these integrals.