In this paper, we use the Yoshida approximation to prove the existence and uniqueness of a solution for the backward doubly stochastic differential equation when the generator is monotone and continuous. Before that we present the results for existence and uniqueness of an adapted solution of the backward doubly stochastic differential equation under some generals conditions.

‎In the present paper‎, ‎optimal heating of temperature field which is modelled as a boundary optimal control problem‎, ‎is investigated in the uncertain environments and then it is solved numerically‎. ‎In physical modelling‎, ‎a partial differential equation with stochastic input and stochastic parameter are applied as the constraint of the optimal control problem‎. ‎Controls are implemented ...

This paper proposes a stochastic finite difference approach, based onhomogenous chaos expansion (SFDHC).The said approach can handle time dependent nonlinear as well as linear systems with deterministic or stochastic initial and boundary conditions. In this approach, included stochastic parameters are modeled as second-order stochastic processes and are expanded using KarhunenLoève expansion, w...

In this note we provide a stochastic method for approximating solutions of ordinary differential equations. To this end, a stochastic variant of the Euler scheme is given by means of Markov chains. For an ordinary differential equation, these approximations are shown to satisfy a Large Number Law, and a Central Limit Theorem for the corresponding fluctuations about the solution of the different...

Stochastic partial differential equations are simply partial differential equations in the presence of uncertainty. Uncertainty, in its simplest form, is modeled by (or taken as) the time derivative (in the sense of distributions) of a Wiener process, known commonly as white noise. The introduction of a random force in a partial differential equation (PDE) arises from the need to explain the fl...

We study finite element approximations of stochastic partial differential equations of Ginzburg-Landau type and the main paradigm considered in this paper is the stochastic Allen-Cahn model. We first demonstrate that the constructed stochastic finite element approximations are within an arbitrary level of tolerance from the corresponding one-dimensional stochastic partial differential equation;...

We consider the stochastic heat equation with a multiplicative colored noise term on Rd for d ≥ 1. First, we prove convergence of a branching particle system in a random environment to this stochastic heat equation with linear noise coefficients. For this stochastic partial differential equation with more general non-Lipschitz noise coefficients we show convergence of associated lattice systems...

We investigate a stochastic optimal control problem where the controlled system is depicted as a stochastic differential delayed equation; however, at the terminal time, the state is constrained in a convex set. We firstly introduce an equivalent backward delayed system depicted as a time-delayed backward stochastic differential equation. Then a stochastic maximum principle is obtained by virtu...

This work focuses on control of surface roughness in sputtering processes including two surface micro-processes, diffusion and erosion. The fluctuation of surface height of such sputtering processes can be described by the stochastic Kuramoto–Sivashinsky equation (KSE), a fourth-order stochastic partial differential equation (PDE). Specifically, we consider sputtering processes, including surfa...