We study Cauchy’s problem for a second-order linear parabolic stochastic partial differential equation (SPDE) driven by a cylindrical Brownian motion. Existence and uniqueness of a generalized (soft) solution is established in Sobolev, Hölder, and Lipschitz classes. We make only minimal assumptions, virtually identical to those common to similar deterministic problems. A stochastic Feynman–Kac ...

In this paper, we show how the Itô-stochastic Magnus expansion can be used to efficiently solve stochastic partial differential equations (SPDE) with two space variables numerically. To end, will first discretize SPDE in only by utilizing finite difference methods and vectorize resulting equation exploiting its sparsity. As a benchmark, apply it case of Langevin constant coefficients, where an ...

This paper presents theoretical advances in the application of Stochastic Partial Differential Equation (SPDE) approach geostatistics. We show a general to construct stationary models related wide class linear SPDEs, with applications spatio-temporal having non-trivial properties. Within framework Generalized Random Fields, criterion for existence and uniqueness solutions this SPDEs is proposed...

Abstract We present a consumption‐based equilibrium framework for credit risk pricing based on the Epstein–Zin (EZ) preferences where default time is modeled as first hitting of boundary and bond investors have imperfect/partial information about firm value. The imperfect generated by underlying observed state variables noisy observation process In addition, consumption, volatility, value are t...

For a superprocess under a stochastic flow, we prove that it has a density with respect to the Lebesgue measure for d = 1 and is singular for d > 1. For d = 1, a stochastic partial differential equation is derived for the density. The regularity of the solution is then proved by using Krylov’s Lp-theory for linear SPDE. A snake representation for this superprocess is established. As application...

Stochastically perturbed Korteweg-de Vries (KdV) equations are widely used to describe the effect of random perturbations on coherent solitary waves. We present a collective coordinate approach waves in stochastically KdV equations. The allows one reduce infinite-dimensional stochastic partial differential equation (SPDE) finite-dimensional for amplitude, width and location wave. reduction prov...

This article is a sequel to [M.Z.Z.1] aimed at completing the characterization of the pathwise local structure of solutions of semi-linear stochastic evolution equations (see’s) and stochastic partial differential equations (spde’s) near stationary solutions. The characterization is expressed in terms of the almost sure long-time behavior of trajectories of the equation in relation to the stati...

Abstract We study the numerical approximation of invariant measure a viscous scalar conservation law, one-dimensional and periodic in space variable stochastically forced with white-in-time but spatially correlated noise. The flux function is assumed to be locally Lipschitz continuous have at most polynomial growth. scheme we employ discretizes stochastic partial differential equation (SPDE) ac...

An innnite system of stochastic diierential equations for the locations and weights of a collection of particles is considered. The particles interact through their weighted empirical measure, V , and V is shown to be the unique solution of a nonlinear stochastic partial diierential equation (SPDE). Conditions are given under which the weighted empirical measure has an L 2-density with respect ...

Abstract We consider a stochastic partial differential equation (SPDE) model for chemorepulsion, with non-linear sensitivity on the one-dimensional torus. By establishing an priori estimate independent of initial data, we show that there exists pathwise unique, global solution to SPDE. Furthermore, associated semi-group is Markov and possesses unique invariant measure, supported Hölder–Besov sp...