## The Annals of Probability

- Ann. Probab.
- Volume 46, Number 1 (2018), 126-174.

### Path-dependent equations and viscosity solutions in infinite dimension

Andrea Cosso, Salvatore Federico, Fausto Gozzi, Mauro Rosestolato, and Nizar Touzi

#### 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.

#### Article information

**Source**

Ann. Probab., Volume 46, Number 1 (2018), 126-174.

**Dates**

Received: February 2015

Revised: October 2016

First available in Project Euclid: 5 February 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1517821220

**Digital Object Identifier**

doi:10.1214/17-AOP1181

**Mathematical Reviews number (MathSciNet)**

MR3758728

**Zentralblatt MATH identifier**

06865120

**Subjects**

Primary: 35D40: Viscosity solutions 35R15: Partial differential equations on infinite-dimensional (e.g. function) spaces (= PDE in infinitely many variables) [See also 46Gxx, 58D25] 60H15: Stochastic partial differential equations [See also 35R60] 60H30: Applications of stochastic analysis (to PDE, etc.)

**Keywords**

Viscosity solutions path-dependent stochastic differential equations path-dependent partial differential equations partial differential equations in infinite dimension

#### Citation

Cosso, Andrea; Federico, Salvatore; Gozzi, Fausto; Rosestolato, Mauro; Touzi, Nizar. Path-dependent equations and viscosity solutions in infinite dimension. Ann. Probab. 46 (2018), no. 1, 126--174. doi:10.1214/17-AOP1181. https://projecteuclid.org/euclid.aop/1517821220