Existence of relaxed stochastic optimal control for G-SDEs with controlled jumps

Abstract In this article, we study the relaxed control problem where the admissible controls are measure-valued processes and the state variable is governed by a G-stochastic differential equation (SDEs) driven by a relaxed Poisson measure where the compensator is a product measure. The control variable appears in the drift and in the jump term. We prove that every solution of our SDE associated to a relaxed control can be written as a limit of a sequence of solutions of SDEs associated to strict controls (stability results). In the end, we show the existence of our relaxed control.