A weak version of path-dependent functional Itô calculus
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Publication:1621446
DOI10.1214/17-AOP1250zbMATH Open1451.60054arXiv1707.04972MaRDI QIDQ1621446
Author name not available (Why is that?)
Publication date: 8 November 2018
Published in: (Search for Journal in Brave)
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.
Full work available at URL: https://arxiv.org/abs/1707.04972
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