We investigate stochastic Volterra equations and their limiting laws. The we consider are driven by a Hilbert space valued \Levy noise integration kernels may have non-linear dependence on the current state of process. Our method is based an embedding into functions which allows to represent solution equation as boundary value partial differential equation. first gather abstract results give mo...

We study the following equation \begin{equation*}\frac{\partial u(t,x)}{\partial t}=\Delta u(t,x)+b\bigl(u(t,x)\bigr)+\sigma \dot{W}(t,x),\quad t>0,\end{equation*} where $\sigma $ is a positive constant and $\dot{W}$ space–time white noise. The initial condition $u(0,x)=u_{0}(x)$ assumed to be nonnegative continuous function. first problem on $[0,1]$ with homogeneous Dirichlet boundary conditio...

Recent results about linear partial differential equations of Kolmogorov type are reviewed. They are examined in the context of financial mathematics: specifically, applications to arbitrage valuation, model calibration and estimation of stochastic processes are discussed.

We have introduced a new explicit numerical method, based on a discrete stochastic process, for solving a class of fractional partial differential equations that model reaction subdiffusion. The scheme is derived from the master equations for the evolution of the probability density of a sum of discrete time random walks. We show that the diffusion limit of the master equations recovers the fra...

In this paper, we present a comparative study between the modified variational iteration method (MVIM) and a hybrid of Fourier transform and variational iteration method (FTVIM). The study outlines the efficiencyand convergence of the two methods. The analysis is illustrated by investigating four singular partial differential equations with variable coefficients. The solution of singular partia...

It is known that if a stochastic process is a solution to a classical Itô stochastic differential equation (SDE), then its transition probabilities satisfy in the weak sense the associated Cauchy problem for the forward Kolmogorov equation. The forward Kolmogorov equation is a parabolic partial differential equation with coefficients determined by the corresponding SDE. Stochastic processes whi...

The purpose of this paper is to present a universal approach to the study of controllability/observability problems for infinite dimensional systems governed by some stochastic/deterministic partial differential equations. The crucial analytic tool is a class of fundamental weighted identities for stochastic/deterministic partial differential operators, via which one can derive the desired glob...