The main objective of this paper and the accompanying one [12] is to provide a notion of viscosity solutions for fully nonlinear parabolic path-dependent PDEs. Our definition extends our previous work [10], focused on the semilinear case, and is crucially based on the nonlinear optimal stopping problem analyzed in [11]. We prove that our notion of viscosity solutions is consistent with the corresponding notion of classical solutions, and satisfies a stability property and a partial comparison result. The latter is a key step for the wellposedness results established in [12]. We also show that the value processes of path-dependent stochastic control problems are viscosity solutions of the corresponding path-dependent dynamic programming equations.

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 It\^{o} calculus recently introduced by Dupire [Functional It\^{o} calculus (2009) Preprint].Comment: Published in at http://dx.doi.org/10.1214/12-AOP788 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

In our previous paper [Ekren, Touzi and Zhang (2015)], we introduced a notion of viscosity solutions for fully nonlinear pathdependent 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 pathdependent PDE has a unique viscosity solution. Uniqueness is implied by a comparison result.

Let X be a bounded càdlàg process with positive jumps defined on the canonical space of continuous paths. We consider the problem of optimal stopping the process X under a nonlinear expectation operator E defined as the supremum of expectations over a weakly compact family of nondominated measures. We introduce the corresponding nonlinear Snell envelope. Our main objective is to extend the Snell envelope characterization to the present context. Namely, we prove that the nonlinear Snell envelope is an E−supermartingale, and an E−martingale up to its first hitting time of the obstacle X. This result is obtained under an additional uniform continuity property of X. We also extend the result in the context of a random horizon optimal stopping problem.This result is crucial for the newly developed theory of viscosity solutions of pathdependent PDEs as introduced in [5], in the semilinear case, and extended to the fully nonlinear case in the accompanying papers [6,7].

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