Optimization problem under change of regime of interest rate

In this paper, we study the problem of maximization of the expected value of the sum of the utility of the terminal wealth and the utility of the consumption, in a case where some sudden jumps in the risk-free interest rate create incompleteness. To solve the problem we use the dual approach. We characterize the value function of the dual problem by a BSDE and the duality between the primal and the dual value functions is exploited to study the BSDE associated to the primal problem.