Skorokhod and pathwise stochastic calculus with respect to an \(L^2\) process (Q2722256)

Given a smooth non-adapted process \(u\), the Skorokhod integral process associated to \(u\) is defined as NEWLINE\[NEWLINEX(t)=\delta(u(\cdot)1_{[0,t]}(\cdot)), \quad t\in [0,1],NEWLINE\]NEWLINE where \(\delta\) stands for the Skorokhod (or Hitsuda-Skorokhod) integral operator. When \(u\) is adapted, the Skorokhod integral process coincides with the classical Itô's integral over Brownian motion. The authors consider a general class of processes of the form \(X(t)=\delta(u(\cdot,t))\), \(t\in [0,1]\), where \(u(\cdot,t)\) belongs (for a fixed \(t\)) to the domain of \(\delta.\) This formulation is more general than the previous one and it is not restrictive. The main tool of this approach is the definition of a Skorokhod type integral operator that acts with respect to \(X(t).\) Under regularity assumptions on \(X(t)\) an anticipating Itô's formula is obtained with sufficient conditions for the existence of quadratic variations and pathwise integrals with respect to \(X(t).\) This result is also stated in the multidimensional case.