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

We propose notions of minimax and viscosity solutions for a class of fully nonlinear path-dependent PDEs with nonlinear, monotone, and coercive operators on Hilbert space. Our main result is well-posedness (existence, uniqueness, and stability) for minimax solutions. A particular novelty is a suitable combination of minimax and viscosity solution techniques in the proof of the comparison principle. One of the main difficulties, the lack of compactness in infinite-dimensional Hilbert spaces, is circumvented by working with suitable compact subsets of our path space. As an application, our theory makes it possible to employ the dynamic programming approach to study optimal control problems for a fairly general class of (delay) evolution equations in the variational framework. Furthermore, differential games associated to such evolution equations can be investigated following the Krasovskiȋ-Subbotin approach similarly as in finite dimensions. 8. Applications to differential games 33 Appendix A. Properties of solution sets of evolution equations 40 Appendix B. Other notions of solutions 45 B.1. Classical solutions 45 B.2. Viscosity solutions 47 References 48 C(S) := {u : S → R continuous with respect to d ∞ }, USC(S) := {u : S → R upper semi-continuous with respect to d ∞ }, LSC(S) := {u : S → R lower semi-continuous with respect to d ∞ }.Clearly, the elements of those function spaces are non-anticipating.2.2. The operator A, related trajectory spaces, and evolution equations. Throughout this work, we fix an operator A : [0, T ] × V → V * and assume that the following hypotheses hold. H(A): (i) The mappings t → A(t, x), v , v ∈ V , are measurable. (ii) Monotonicity: For a.e. t ∈ (0, T ) and every x, y ∈ V , A(t, x) − A(t, y), x − y ≥ 0. (iii) Hemicontinuity: For a.e. t ∈ (0, T ) and every x, y, v ∈ V , the mappings s → A(t, x + sy), v , [0, 1] → R, are continuous.

This paper introduces the path derivatives, in the spirit of Dupire's functional Itô calculus, for the controlled paths in the rough path theory with possibly non-geometric rough paths. The theory allows us to deal with rough integration and rough PDEs in the same manner as standard stochastic calculus. We next study rough PDEs with coefficients depending on the rough path itself, which corresponds to stochastic PDEs with random coefficients. Such coefficients is less regular in the time variable and is not covered in the existing literature. The results are useful for studying viscosity solutions of stochastic PDEs.

Abstract. We extend the notion of viscosity solutions for path-dependent PDEs introduced by Ekren et al. [Ann. Probab. 42 (2014), no. 1, 204-236] to path-dependent integro-differential equations and establish well-posedness, i.e., existence, uniqueness, and stability, for a class of semilinear path-dependent integro-differential equations with uniformly continuous data. Closely related are non-Markovian backward SDEs with jumps, which provide a probabilistic representation for solutions of our equations. The results are potentially useful for applications using non-Markovian jump-diffusion models. IntroductionThe goal of this paper is to extend the theory of viscosity solutions (in the sense of [17] and [18]) for path-dependent partial differential equations (PPDEs) to path-dependent integro-differential equations. In particular, we investigate semilinear path-dependent integro-differential equations of the form and I is an integral operator of the formWell-posedness for semilinear PPDEs has been first established by Ekren, Keller, Touzi, and Zhang [17], where also the here used notion of viscosity solutions has been introduced. [40]) motivated by the study of differential games. In the case of PDEs of first order, minimax and viscosity solutions are equivalent (see [41]). Another approach for generalized solution for first-order PPDEs can be found in work by Aubin and Haddad [1], where so-called Clio derivatives for path-dependent functionals are introduced in order to study certain path-dependent Hamiltion-Jacobi-Bellman equations that occur in portfolio theory.Possible applications of path-dependent integro-differential equations are non-Markovian problems in control, differential games, and financial mathematics that involve jump processes.Some comments about differences between PDEs and PPDEs seem to be in order. Contrary to standard PDEs, even linear PPDEs have rarely classical solutions in most relevant situations. Hence, one needs to consider a weaker forms of solutions. In the case of PDEs, the notion of viscosity solutions introduced by Crandall and Lions [11] turned out to be extremely successful. The main difficulty in the path-dependent case compared to the standard PDE case is the lack of local compactness of the state space, e.g.,Local compactness is essential for proofs of uniqueness of viscosity solutions to PDEs, i.e., PDE standard methods can, in general, not easily adapted to the path-dependent case. The main contribution of [17] was to replace the pointwise supremum/ infimum occuring in the definition of viscosity solutions to PDEs via test functions by an optimal stopping problem. The lack of local compactness could be circumvented by the existence of an optimal stopping time. This is crucial in establishing PPIDE 3 the comparison principle. In this paper, additional intricacies caused by the jumps have to be faced. For example, it turns out that in contrast to the PPDE case the uniform topology is not always appropriate. In order to prove the comparison principle, it seems necessary t...

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