Annals of Probability

On viscosity solutions of path dependent PDEs

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

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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


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References

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