Viscosity solutions of path-dependent integro-differential equations

Abstract

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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