Clark-Ocone type formula for non-semimartingales with finite quadratic variation

TL;DR

This paper develops a Clark-Ocone type formula for processes with finite quadratic variation in Banach spaces, extending stochastic calculus tools to non-semimartingales with memory effects.

Contribution

It introduces a framework for finite quadratic variation processes in Banach spaces and derives a generalized Clark-Ocone formula using infinite-dimensional PDEs.

Findings

Established a new Itô formula for Banach space-valued processes.

Derived a Clark-Ocone representation for non-semimartingales with Brownian-like quadratic variation.

Connected stochastic calculus with infinite-dimensional PDE solutions.

Abstract

We provide a suitable framework for the concept of finite quadratic variation for processes with values in a separable Banach space $B$ using the language of stochastic calculus via regularizations, introduced in the case $B= \R$ by the second author and P. Vallois. To a real continuous process $X$ we associate the Banach valued process $X(\cdot)$, called {\it window} process, which describes the evolution of $X$ taking into account a memory $\tau>0$. The natural state space for $X(\cdot)$ is the Banach space of continuous functions on $[-\tau,0]$. If $X$ is a real finite quadratic variation process, an appropriated It\^o formula is presented, from which we derive a generalized Clark-Ocone formula for non-semimartingales having the same quadratic variation as Brownian motion. The representation is based on solutions of an infinite dimensional PDE.