Weak Functional It\^o Calculus and Applications

We introduce a variational theory for processes adapted to the multi-dimensional Brownian motion filtration. The theory provides a differential structure which describes the infinitesimal evolution of Wiener functionals at very small scales. The Markov property is replaced by a finite-dimensional approximation procedure based on controlled inter-arrival times and jumps of approximating martingales. The theory reveals that a large class of adapted processes follow a differential rule which is similar in nature to a fundamental theorem of calculus in the context of Wiener functionals. Null stochastic derivative term turns out to be a non-Markovian version of the classical second order parabolic operator and connections with theory of local-times and $(p,q)$-variation regularity are established. The framework extends the pathwise functional calculus and it opens the way to obtain variational principles for processes. Applications to semi-linear variational equations and stochastic variational inequalities are presented. In particular, we provide a feasible dynamic programming principle for fully non-Markovian/non-semimartingale optimal stopping problems.