Viscosity Solutions of Path-dependent Integro-differential Equations

TL;DR

This paper extends the concept of viscosity solutions to path-dependent integro-differential equations, establishing their well-posedness and linking them to non-Markovian jump SDEs, with potential applications in jump-diffusion models.

Contribution

It introduces a new framework for viscosity solutions of path-dependent integro-differential equations, proving existence, uniqueness, and stability, and connects these solutions to non-Markovian backward SDEs with jumps.

Findings

Established well-posedness for a class of path-dependent integro-differential equations.

Linked solutions to non-Markovian backward SDEs with jumps.

Potential applications in non-Markovian jump-diffusion models.

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.