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In this paper, we study the classical problem of maximization of the sum ofthe utility of the terminal wealth and the utility of the consumption, in acase 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 andthe duality between the primal and the dual value functions is exploited tostudy the BSDE associated to the primal problem.

Optimization problem under change of regime of interest rate