On viscosity solutions of path dependent PDES

Ibrahim Ekren; Christian Keller; Nizar Touzi; Jianfeng Zhang

doi:10.1214/12-AOP788

On viscosity solutions of path dependent PDES

Ibrahim Ekren, Christian Keller, Nizar Touzi, Jianfeng Zhang

Finance and Risk Engineering

Research output: Contribution to journal › Article › peer-review

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].

Original language	English (US)
Pages (from-to)	204-236
Number of pages	33
Journal	Annals of Probability
Volume	42
Issue number	1
DOIs	https://doi.org/10.1214/12-AOP788
State	Published - Jan 2014

Keywords

Backward SDEs
Comparison principle
Functional itô formula
Path dependent PDEs
Viscosity solutions

ASJC Scopus subject areas

Statistics and Probability
Statistics, Probability and Uncertainty

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10.1214/12-AOP788

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AU - Touzi, Nizar

AU - Zhang, Jianfeng

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AB - 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].

KW - Backward SDEs

KW - Comparison principle

KW - Functional itô formula

KW - Path dependent PDEs

KW - Viscosity solutions

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On viscosity solutions of path dependent PDES

Abstract

Keywords

ASJC Scopus subject areas

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