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November 2018 A weak version of path-dependent functional Itô calculus
Dorival Leão, Alberto Ohashi, Alexandre B. Simas
Ann. Probab. 46(6): 3399-3441 (November 2018). DOI: 10.1214/17-AOP1250

Abstract

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.

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Dorival Leão. Alberto Ohashi. Alexandre B. Simas. "A weak version of path-dependent functional Itô calculus." Ann. Probab. 46 (6) 3399 - 3441, November 2018. https://doi.org/10.1214/17-AOP1250

Information

Received: 1 November 2015; Revised: 1 November 2017; Published: November 2018
First available in Project Euclid: 25 September 2018

zbMATH: 06975490
MathSciNet: MR3857859
Digital Object Identifier: 10.1214/17-AOP1250

Subjects:
Primary: 60H07
Secondary: 60H25

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 6 • November 2018
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