The Annals of Probability
- Ann. Probab.
- Volume 44, Number 4 (2016), 2507-2553.
Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II
In our previous paper [Ekren, Touzi and Zhang (2015)], we introduced a notion of viscosity solutions for fully nonlinear path-dependent PDEs, extending the semilinear case of Ekren et al. [Ann. Probab. 42 (2014) 204–236], which satisfies a partial comparison result under standard Lipshitz-type assumptions. The main result of this paper provides a full, well-posedness result under an additional assumption, formulated on some partial differential equation, defined locally by freezing the path. Namely, assuming further that such path-frozen standard PDEs satisfy the comparison principle and the Perron approach for existence, we prove that the nonlinear path-dependent PDE has a unique viscosity solution. Uniqueness is implied by a comparison result.
Ann. Probab., Volume 44, Number 4 (2016), 2507-2553.
Received: May 2013
Revised: September 2014
First available in Project Euclid: 2 August 2016
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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.)
Ekren, Ibrahim; Touzi, Nizar; Zhang, Jianfeng. Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II. Ann. Probab. 44 (2016), no. 4, 2507--2553. doi:10.1214/15-AOP1027. https://projecteuclid.org/euclid.aop/1470139148