An explicit representation formula for a solution is given in Theorem 1, when g is a bounded smooth vector field. The case of a complete vector field g ∈ C1(Rn,Rn) is analyzed in Theorem 2, introducing adequate stopping times. The main support in writing a solution comes from the solution yλ(τ1, τ2) = G (F (τ1, τ2)) [λ], (τ1, τ2) ∈ R, λ ∈ R, satisfying a deterministic gradient system  ∂τ1yλ (...

This paper deals with robust exponential stability of Markovian jumping stochastic Cohen–Grossberg neural networks (MJSCGNNs) with mode-dependent probabilistic time-varying delays, continuously distributed delays and impulsive perturbations. By construction of novel Lyapunov–Krasovskii functional having the triple integral terms, the double integral terms having the positive definite matrices d...

Cylindrical Wiener processes in real separable Banach spaces are defined, and an approximation theorem involving scalar Wiener processes is given for such processes. A weak stochastic integral for Banach spaces involving a cylindrical Wiener process as integrator and an operator-valued stochastic process as integrand is defined. Basic properties of this integral are stated and proved. A class o...

We consider the solution u(x, t) to a stochastic heat equation. For fixed x, the process F (t) = u(x, t) has a nontrivial quartic variation. It follows that F is not a semimartingale, so a stochastic integral with respect to F cannot be defined in the classical Itô sense. We show that for sufficiently differentiable functions g, a stochastic integral ∫ g(F ) dF exists as a limit in distribution...

The paper is devoted to the existence of integral functionals $\int_0^\infty f(X(t))\,{\mathrm{d}t}$ for several classes processes in $\mathbb{R}$ with $d\ge 3$. Some examples such as Brownian motion, fractional compound Poisson process, Markov admitting densities transitional probabilities are considered.