Optimal stopping and stochastic control differential games for jump diffusions

We study stochastic differential games of jump diffusions driven by Brownian motions and compensated Poisson random measures, where one of the players can choose the stochastic control and the other player can decide when to stop the system. We prove a verification theorem for such games in terms of a Hamilton–Jacobi–Bellman variational inequality. The results are applied to study some specific examples, including optimal resource extraction in a worst-case scenario, and risk minimizing optimal portfolio and stopping. This is an Accepted Manuscript of an article published by Taylor & Francis in Stochastics An International Journal of Probability and Stochastic Processes: formerly Stochastics and Stochastics Report