Pathwise Taylor Expansions for Itô Random Fields

Abstract. In this paper we study the pathwise stochastic Taylor expansion, in the sense of our previous work [3], for a class of Itô-type random fields in which the di↵usion part is allowed to contain both the random field itself and its spatial derivatives. Random fields of such an “self-exciting” type particularly contains the fully nonlinear stochastic PDEs of curvature driven di↵usion, as well as certain stochastic Hamilton-Jacobi-Bellman equations. We introduce the new notion of “n-fold” derivatives of a random field, as a fundamental device to cope with the special self-exciting nature. Unlike our previous work [3], our new expansion can be defined around any random time-space point (⌧, ⇠), where the temporal component ⌧ does not even have to be a stopping time. Moreover, the exceptional null set is independent of the choice of the random point (⌧, ⇠). As an application, we show how this new form of pathwise Taylor expansion could lead to a di↵erent treatment of the stochastic characteristics for a class of fully nonlinear SPDEs whose di↵usion term involves both the solution and its gradient, and hence lead to a definition of the stochastic viscosity solution for such SPDEs, which is new in the literature, and potentially of essential importance in stochastic control theory.