Before revision:  We assume that the Brownian motion and jump process are independent, and there exists a  probability density function for the jump times and marks . Under these assumptions, we decompose the global controller-stopper problem into controller-stopper problems with respect to the Brownian filtration, which are determined by a backward induction.
After revision:  We assume that the Brownian motion and jump process are independent, and there exists a probability density function for the jump times and marks given the filtration of the Brownian motion and decompose the global controller-stopper problem into controller-stopper problems with respect to the Brownian filtration, which are determined by a backward induction.