## The Annals of Probability

- Ann. Probab.
- Volume 42, Number 1 (2014), 204-236.

### On viscosity solutions of path dependent PDEs

Ibrahim Ekren, Christian Keller, Nizar Touzi, and Jianfeng Zhang

#### Abstract

In this paper we propose a notion of viscosity solutions for path dependent semi-linear parabolic PDEs. This can also be viewed as viscosity solutions of non-Markovian backward SDEs, and thus extends the well-known nonlinear Feynman–Kac formula to non-Markovian case. We shall prove the existence, uniqueness, stability and comparison principle for the viscosity solutions. The key ingredient of our approach is a functional Itô calculus recently introduced by Dupire [Functional Itô calculus (2009) Preprint].

#### Article information

**Source**

Ann. Probab., Volume 42, Number 1 (2014), 204-236.

**Dates**

First available in Project Euclid: 9 January 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1389278524

**Digital Object Identifier**

doi:10.1214/12-AOP788

**Mathematical Reviews number (MathSciNet)**

MR3161485

**Zentralblatt MATH identifier**

1320.35154

**Subjects**

Primary: 35D40: Viscosity solutions 35K10: Second-order parabolic equations 60H10: Stochastic ordinary differential equations [See also 34F05] 60H30: Applications of stochastic analysis (to PDE, etc.)

**Keywords**

Path dependent PDEs backward SDEs functional Itô formula viscosity solutions comparison principle

#### Citation

Ekren, Ibrahim; Keller, Christian; Touzi, Nizar; Zhang, Jianfeng. On viscosity solutions of path dependent PDEs. Ann. Probab. 42 (2014), no. 1, 204--236. doi:10.1214/12-AOP788. https://projecteuclid.org/euclid.aop/1389278524