A WEAK VERSION OF PATH-DEPENDENT FUNCTIONAL ITO CALCULUS

• 成果类型: Article
• 署名作者: Leao, Dorival; Ohashi, Alberto; Simas, Alexandre B.
• 署名单位: Universidade de Sao Paulo; Universidade Federal da Paraiba
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/17-AOP1250
• 发表日期: 2018
• 页码: 3399-3441
• 关键词: viscosity solutions; brownian-motion; pdes; martingales; CONVERGENCE; EQUATIONS; formulas

摘要

We introduce a variational theory for processes adapted to the multidimensional 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 generator, 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.