Stochastic viscosity solutions for nonlinear stochastic partial differential equations. I
type: Item
instance of: scholarly article
title: Stochastic viscosity solutions for nonlinear stochastic partial differential equations. I
zbMATH Open document ID: 1053.60065
DOI: 10.1016/S0304-4149(00)00093-4
author: Jin Ma
author: Rainer Buckdahn
published in: Stochastic Processes and their Applications
publication date: 2004-11-26T00:00:00Z
review text: A nonlinear stochastic partial differential equation in the Stratonovich form \[ du = \bigl \{ Au + f(t,x,u,\sigma ^ *(x)Du)\bigr \}\,dt + \sum ^ {d}_ {i=1} g_ {i}(t,x,u)\circ\,dB^ {i}_ {t}, \quad (t,x)\in (0,T)\times \mathbb R^ {m}, \tag{1} \] with initial condition \(u(0,\cdot ) = u_ 0\) on \(\mathbb R^ {m}\) is studied. In (1), \(B\) denotes the standard \(d\)-dimensional Wiener process and \(A\) is a second-order differential operator \[ A = \sum ^ {m}_ {i,j=1} \sum ^ {k}_ {l=1}\sigma _ {il}(x) \sigma _ {jl}(x)\partial ^ 2_ {x_ {i}x_ {j}} + \sum ^ {m}_ {i=1} \beta _ {i}(x)\partial _ {x_ {i}}. \] It is supposed that \(u_ 0\) is a continuous function of a polynomial growth, \(\sigma \), \(\beta \) are Lipschitz continuous functions, \(g\) is a continuous function sufficiently smooth in both the second and the third variables, and \(f\) is a progressively measurable random field, almost surely uniformly Lipschitz continuous in all variables. By extending the well known deterministic concept, the authors propose a new definition of a stochastic viscosity solution to (1). A transformation converting (1) into a partial differential equation with random coefficient (but with no martingale term) is found such that stochastic viscosity solutions to (1) are mapped in a one-to-one way to pathwise viscosity solutions of this random PDE. By means of this transformation and the theory of backward doubly stochastic differential equations, existence of stochastic viscosity solutions of (1) is established. [For part II see below, Zbl 1053.60064.]
reviewed by: Jan Seidler
Mathematics Subject Classification ID: 60H15
zbMATH DE Number: 2119495
zbMATH Keywords: stochastic partial differential equations
zbMATH Keywords: viscosity solutions
zbMATH Keywords: backward doubly stochastic differential equations
MaRDI profile type: Publication
cites work: Q3997913
cites work: Stochastic viscosity solutions for nonlinear stochastic partial differential equations. II.
cites work: Viscosity Solutions of Hamilton-Jacobi Equations
cites work: User’s guide to viscosity solutions of second order partial differential equations
cites work: Q4131410
cites work: Stochastic differential equations of jump type and Lévy processes in diffeomorphisms group
cites work: Résultats d'existence et d'unicité pour des équations différentielles stochastiques rétrogrades avec des générateurs à croissance quadratique
cites work: Q4039796
cites work: Q3995203
cites work: Fully nonlinear stochastic partial differential equations
cites work: Fully nonlinear stochastic partial differential equations: non-smooth equations and applications
cites work: Stochastic calculus with anticipating integrands
cites work: A generalized Itô-Ventzell formula. Application to a class of anticipating stochastic differential equations
cites work: Q4029028
cites work: Backward doubly stochastic differential equations and systems of quasilinear SPDEs
cites work: Q3996259
cites work: On the gap between deterministic and stochastic ordinary differential equations
full work available at URL: s0304-4149(00)00093-4
OpenAlex ID: W2009802270
description: scientific article; zbMATH DE number 2119495