Optimization problem under change of regime of interest rate

In this paper, we study the classical problem of maximization of the sum of the utility of the terminal wealth and the utility of the consumption, in a case where a sudden jump in the risk-free interest rate creates incompleteness. The value function of the dual problem is proved to be solution of a BSDE and the duality between the primal and the dual value functions is exploited to study the BSDE associated to the primal problem.