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January 2018 Path-dependent equations and viscosity solutions in infinite dimension
Andrea Cosso, Salvatore Federico, Fausto Gozzi, Mauro Rosestolato, Nizar Touzi
Ann. Probab. 46(1): 126-174 (January 2018). DOI: 10.1214/17-AOP1181

Abstract

Path-dependent partial differential equations (PPDEs) are natural objects to study when one deals with non-Markovian models. Recently, after the introduction of the so-called pathwise (or functional or Dupire) calculus [see Dupire (2009)], in the case of finite-dimensional underlying space various papers have been devoted to studying the well-posedness of such kind of equations, both from the point of view of regular solutions [see, e.g., Dupire (2009) and Cont (2016) Stochastic Integration by Parts and Functional Itô Calculus 115–207, Birkhäuser] and viscosity solutions [see, e.g., Ekren et al. (2014) Ann. Probab. 42 204–236]. In this paper, motivated by the study of models driven by path-dependent stochastic PDEs, we give a first well-posedness result for viscosity solutions of PPDEs when the underlying space is a separable Hilbert space. We also observe that, in contrast with the finite-dimensional case, our well-posedness result, even in the Markovian case, applies to equations which cannot be treated, up to now, with the known theory of viscosity solutions.

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Andrea Cosso. Salvatore Federico. Fausto Gozzi. Mauro Rosestolato. Nizar Touzi. "Path-dependent equations and viscosity solutions in infinite dimension." Ann. Probab. 46 (1) 126 - 174, January 2018. https://doi.org/10.1214/17-AOP1181

Information

Received: 1 February 2015; Revised: 1 October 2016; Published: January 2018
First available in Project Euclid: 5 February 2018

zbMATH: 06865120
MathSciNet: MR3758728
Digital Object Identifier: 10.1214/17-AOP1181

Subjects:
Primary: 35D40 , 35R15 , 60H15 , 60H30

Keywords: partial differential equations in infinite dimension , path-dependent partial differential equations , path-dependent stochastic differential equations , viscosity solutions

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 1 • January 2018
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