# Stochastic viscosity solutions for stochastic integral-partial differential equations and singular stochastic control

## TL;DR

This paper develops a framework for stochastic viscosity solutions of semilinear SIPDEs, introduces a new class of GBDSDEs driven by Brownian motions and Poisson measures, and applies these to stochastic control problems.

## Contribution

It introduces a novel class of GBDSDEs with jumps, defines stochastic viscosity solutions for SIPDEs, and establishes maximum principles for stochastic control.

## Key findings

- Proved existence and uniqueness of solutions for the new GBDSDEs.
- Provided a probabilistic representation for stochastic viscosity solutions.
- Established stochastic maximum principles for control systems modeled by GBDSDEs.

## Abstract

In this article, we mainly study stochastic viscosity solutions for a class of semilinear stochastic integral-partial differential equations (SIPDEs). We investigate a new class of generalized backward doubly stochastic differential equations (GBDSDEs) driven by two independent Brownian motions and an independent Poisson random measure, which involves an integral with respect to a c\`{a}dl\`{a}g increasing process. We first derive existence and uniqueness of the solution of GBDSDEs with general jumps. We then introduce the definition of stochastic viscosity solutions of SIPDEs and give a probabilistic representation for stochastic viscosity solutions of semilinear SIPDEs with nonlinear Neumann boundary conditions. Finally, we establish stochastic maximum principles for the optimal control of a stochastic system modelled by a GBDSDE with general jumps.