  In this paper we study the {\it pathwise stochastic Taylor expansion}, in the
sense of our previous work \cite{Buckdahn_Ma_02}, for a class of It\^o-type
random fields in which the diffusion 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 diffusion, 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 \cite{Buckdahn_Ma_02}, our new
expansion can be defined around any random time-space point $(\t,\xi)$, where
the temporal component $\t$ does not even have to be a stopping time. Moreover,
the exceptional null set is independent of the choice of the random point
$(\t,\xi)$. As an application, we show how this new form of pathwise Taylor
expansion could lead to a different treatment of the stochastic characteristics
for a class of fully nonlinear SPDEs whose diffusion term involves both the
solution and its gradient, and hence lead to a definition of the {\it
stochastic viscosity solution} for such SPDEs, which is new in the literature.
