The Annals of Probability
- Ann. Probab.
- Volume 46, Number 6 (2018), 3399-3441.
A weak version of path-dependent functional Itô calculus
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.
Ann. Probab., Volume 46, Number 6 (2018), 3399-3441.
Received: November 2015
Revised: November 2017
First available in Project Euclid: 25 September 2018
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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
- 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 .