  We consider controller-stopper problems in which the controlled processes can
have jumps. The global filtration is represented by the Brownian filtration,
enlarged by the filtration generated by the jump process. We assume that there
exists a conditional 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. We apply our
decomposition method to indifference pricing of American options under multiple
default risk. The backward induction leads to a system of reflected backward
stochastic differential equations (RBSDEs). We show that there exists a
solution to this RBSDE system and that the solution provides a characterization
of the value function.
