We derive the hydrodynamic limit of a kinetic equation with stochastic, short range perturbation velocity operator. Under some mixing hypotheses on stochastic perturbation, we establish diffusion-approximation result: obtain is parabolic partial differential macroscopic parameter, density here.

We focus on the use of two stable and accurate explicit finite difference schemes in order to approximate the solution of stochastic partial differential equations of It¨o type, in particular, parabolic equations. The main properties of these deterministic difference methods, i.e., convergence, consistency, and stability, are separately developed for the stochastic cases.

An effective macroscopic model for a stochastic microscopic system is derived. The original microscopic system is modeled by a stochastic partial differential equation defined on a domain perforated with small holes or heterogeneities. The homogenized effective model is still a stochastic partial differential equation but defined on a unified domain without holes. The solutions of the microscop...

Stochastic differential equations (SDE) are used to model many situations including population dynamics, protein kinetics, turbulence, finance, and engineering [5, 6, 1]. Knowing the solution of the SDE in question leads to interesting analysis of the trajectories. Most SDE are unsolvable analytically and other methods must be used to analyze properties of the stochastic process. From the SDE, ...

In this paper we study a class of pathwise stochastic control problems in which the optimality is allowed to depend on the paths of exogenous noise (or information). Such a phenomenon can be illustrated by considering a particular investor who wants to take advantage of certain extra information but in a completely legal manner. We show that such a control problem may not even have a “minimizin...

In this paper, we consider learning of spatio-temporal processes by formulating a Gaussian process model as a solution to an evolution type stochastic partial differential equation. Our approach is based on converting the stochastic infinite-dimensional differential equation into a finite dimensional linear time invariant (LTI) stochastic differential equation (SDE) by discretizing the process ...

Lagrange multiplier methods are standard fare in elementary calculus courses, and they play a central role in economic applications of calculus because they often turn out to have interpretations as prices or shadow prices. You have seen them generalized to cover dynamic, non-stochastic models as Hamiltonian methods, or as byproducts of using Pontryagin’s maximum principle. In static models Lag...

Lagrange multiplier methods are standard fare in elementary calculus courses, and they play a central role in economic applications of calculus because they often turn out to have interpretations as prices or shadow prices. You have seen them generalized to cover dynamic, non-stochastic models as Hamiltonian methods, or as byproducts of using Pontryagin’s maximum principle. In static models Lag...

In this paper we study a class of pathwise stochastic control problems in which the optimality is allowed to depend on the paths of exogenous noise (or information). Such a phenomenon can be illustrated by considering a particular investor who wants to take advantage of certain extra information but in a completely legal manner. We show that such a control problem may not even have a “minimizin...