The Annals of Probability

A weak version of path-dependent functional Itô calculus

Dorival Leão, Alberto Ohashi, and Alexandre B. Simas

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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.

Article information

Source
Ann. Probab., Volume 46, Number 6 (2018), 3399-3441.

Dates
Received: November 2015
Revised: November 2017
First available in Project Euclid: 25 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.aop/1537862437

Digital Object Identifier
doi:10.1214/17-AOP1250

Mathematical Reviews number (MathSciNet)
MR3857859

Zentralblatt MATH identifier
06975490

Subjects
Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus
Secondary: 60H25: Random operators and equations [See also 47B80]

Keywords
Stochastic calculus of variations functional Itô calculus

Citation

Leão, Dorival; Ohashi, Alberto; Simas, Alexandre B. A weak version of path-dependent functional Itô calculus. Ann. Probab. 46 (2018), no. 6, 3399--3441. doi:10.1214/17-AOP1250. https://projecteuclid.org/euclid.aop/1537862437


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Supplemental materials

  • Supplement to “A weak version of path-dependent functional Itô calculus”. The proofs of Lemmas 2.1, 2.3, 3.2, 4.4 and Theorem 4.4 are provided in the online supplement [33].