Infinite dimensional stochastic calculus via regularization

This paper develops some aspects of stochastic calculus via regularization to Banach valued processes. An original concept of $\chi$-quadratic variation is introduced, where $\chi$ is a subspace of the dual of a tensor product $B \otimes B$ where $B$ is the values space of some process $X$ process. Particular interest is devoted to the case when $B$ is the space of real continuous functions defined on $[-\tau,0]$, $\tau>0$. Ito formulae and stability of finite $\chi$-quadratic variation processes are established. Attention is deserved to a finite real quadratic variation (for instance Dirichlet, weak Dirichlet) process $X$. The $C([-\tau,0])$-valued process $X(\cdot)$ defined by $X_t(y) = X_{t+y}$, where $y \in [-\tau,0]$, is called {\it window} process. Let $T >0$. If $X$ is a finite quadratic variation process such that $[X]_t = t$ and $h = H(X_T(\cdot))$ where $H:C([-T,0])\longrightarrow \R$ is $L^{2}([-T,0])$-smooth or $H$ non smooth but finitely based it is possible to represent $h$ as a sum of a real $H_{0}$ plus a forward integral of type $\int_0^T \xi d^-X$ where $H_{0}$ and $\xi$ are explicitly given. This representation result will be strictly linked with a function $u:[0,T]\times C([-T,0])\longrightarrow \R$ which in general solves an infinite dimensional partial differential equation with the property $H_{0}=u(0, X_{0}(\cdot))$, $\xi_{t}=D^{\delta_{0}}u(t, X_{t}(\cdot)):=Du(t,X_{t}(\cdot))(\{0\})$. This decomposition generalizes the Clark-Ocone formula which is true when $X$ is the standard Brownian motion $W$. The financial perspective of this work is related to hedging theory of path dependent options without semimartingales.