Pathwise Taylor expansions for random fields on multiple dimensional paths

In this paper we establish the pathwise Taylor expansions for random fields that are “regular” in terms of Dupire’s path-derivatives [6]. Using the language of pathwise calculus, we carry out the Taylor expansion naturally to any order and for any dimension, which extends the result of Buckdahn et al. (2011). More importantly, the expansion can be both “forward” and “backward”, and the remainder is estimated in a pathwise manner. This result will be the main building block for our new notion of viscosity solution to forward path-dependent PDEs corresponding to (forward) stochastic PDEs in our accompanying paper Buckdahn et al. [4].