Stochastic integrals of anticipating processes and predictable dual projections (Q1301341)

Let \(W\) be a standard Brownian motion and \(({\mathcal F}_t)\) the filtration generated by \(W\). In difference to the (anticipating) Skorokhod integral \(\delta\) defined for \(u\in L^2({\mathcal F}_1 \otimes {\mathcal B}([0,1]), dsdP)\) for which there exists a \(\delta(u)\in L^2({\mathcal F}_1,P)\) such that \(E [F\delta (u)]=E [\int^1_0D_tFu_tdt]\), \(\forall F\in D^{1,2}\), the authors introduce an anticipating integral \(\delta^\mu\) associated to a (nonadapted) increasing process \((\mu_t)\) with dual predictable projection \(t\) defined by the relation \[ E\bigl[F \delta^\mu (u)\bigr]= E\left[ \int^1_0 E\bigl[ D_tF\mid {\mathcal F}_t\bigr] u_td\mu_t \right],\;\forall F. \] This integral \(\delta^\mu (u)\) coincides with the Itô integral of the projection \(u^\mu\) of \[ u\in L^2({\mathcal F}_1\oplus {\mathcal B}([0,1]), d\mu_sdP) \] onto its subspace of predictable processes. For \(d\mu_t:= -d(\sup_{s\in[0,t]}(W_s-W_1))^2\) the projection \(u\to u^\mu\) is studied. In particular, it is characterized via a desintegration formula for \(\mu\) involving the local time of \(W\).