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 processe...
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| Veröffentlicht in: | The Annals of probability 2018-11, Vol.46 (6), p.3399-3441 |
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| container_title | The Annals of probability |
| container_volume | 46 |
| creator | Leão, Dorival Ohashi, Alberto Simas, Alexandre B. |
| description | 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 ofWiener 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. |
| doi_str_mv | 10.1214/17-AOP1250 |
| format | Article |
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| ispartof | The Annals of probability, 2018-11, Vol.46 (6), p.3399-3441 |
| issn | 0091-1798 2168-894X |
| language | eng |
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| source | JSTOR Mathematics & Statistics; Jstor Complete Legacy; EZB-FREE-00999 freely available EZB journals; Project Euclid Complete |
| title | A WEAK VERSION OF PATH-DEPENDENT FUNCTIONAL ITÔ CALCULUS |
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