Pathwise stochastic Taylor expansions and stochastic viscosity solutions for fully nonlinear stochastic PDEs

• 成果类型: Article
• 署名作者: Buckdahn, R; Ma, J
• 署名单位: Universite de Bretagne Occidentale; Purdue University System; Purdue University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2002
• 页码: 1131-1171
• 关键词: partial-differential equations

摘要

In this paper we study a new type of Taylor expansion for Ito-type random fields, up to the second order. We show that an M-type random field with reasonably regular integrands can be expanded, up to the second order, to the linear combination of increments of temporal and spatial variables, as well as the driven Brownian motion, around even a random (t, x)-point. Also, the remainder can be estimated in a pathwise manner. We then show that such a Taylor expansion is valid for the solutions to a fairly large class of stochastic differential equations with parameters, or even fully-nonlinear stochastic partial differential equations, whenever they exist. Using such analysis we then propose a new definition of stochastic viscosity solution for fully nonlinear stochastic PDEs, in the spirit of its deterministic counterpart. We prove that this new definition is actually equivalent to the one proposed in our previous works [2] and [3], at least for a class of quasilinear SPDEs.