Before revision:  We separate the latter into a pre-default and a post-default dynamic optimization subproblems, and obtain two coupled Hamilton-Jacobi-Bellman equations for the optimal value functions. We obtain a complete solution to the post-default optimization subproblem, and prove a verification theorem for the solution of the pre-default optimization subproblem .
After revision:  We separate the latter into a pre-default and a post-default dynamic optimization subproblems, and obtain two coupled Hamilton-Jacobi-Bellman (HJB) partial differential equations. We prove existence and uniqueness of a globally bounded classical solution to the post-default optimization subproblem, and prove a verification theorem for the solution of the pre-default optimization subproblem .