A weak version of path-dependent functional It\^o calculus

TL;DR

This paper develops a variational framework for analyzing Wiener functionals using a weak path-dependent calculus, enabling the computation of sensitivities and infinitesimal generators through finite-dimensional approximations.

Contribution

It introduces a novel weak path-dependent calculus for Wiener functionals, extending the differential analysis beyond semimartingales via finite-dimensional approximations.

Findings

Provides convergence results for the approximation scheme.

Shows solutions of BSDEs as energy minimizers in this framework.

Enables pathwise computation of sensitivities and generators.

Abstract

We introduce a variational theory for processes adapted to the multi-dimensional Brownian motion filtration that provides a differential structure allowing to describe infinitesimal evolution of Wiener functionals at very small scales. The main novel idea is to compute the "sensitivities" of processes, namely derivatives of martingale components and a weak notion of infinitesimal generators, via a finite-dimensional approximation procedure based on controlled inter-arrival times and approximating martingales. The theory comes with convergence results that allow to interpret a large class of Wiener functionals beyond semimartingales as limiting objects of differential forms which can be computed path wisely over finite-dimensional spaces. The theory reveals that solutions of BSDEs are minimizers of energy functionals w.r.t Brownian motion driving noise.