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

Abstract

In this paper we study a new type of "Taylor expansion" for Itô-type random fields, up to the second order. We show that an Itô-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, at least for a class of quasilinear SPDEs.