The Clark-Ocone formula for vector valued Wiener functionals (Q2577512)

Consider an abstract Wiener space \((W,H,\mu)\) and the filtration induced by a strictly increasing continuous resolution of the identity on \(H\). Let \(v\) be a random variable defined on \(W\) and with values in a Banach space. It is proved that if \(v\) is in a suitable Sobolev space, then it has a Clark-Ocone representation of the form \(v=E\,v+\delta(\Pi\nabla v).\) In this formula, \(\delta\) is a divergence operator replacing the Itô integration (which is not always well defined in this framework), and the process \(K=\Pi\nabla v\) is weakly adapted. This result is then applied to \(W\)-valued random variables of the form \(w\mapsto Tw\), where \(T\) is measure preserving.