Summary: In this paper, we study the exponential utility maximization problem in an incomplete market with a default time inducing a discontinuity in the price of stocks We consider the case of strategies valued in a closed set. Using dynamic programming and Backward Stochastic Differential Equation (BSDE) techniques, we provide a characterization of the value function as the maximal subsolution of a BSDE and an optimality criterium. Moreover, in the case of bounded coefficients, the value function is shown to be the maximal solution of a BSDE. Moreover, the value function can be written as the limit of a sequence of processes which can be characterized as the solutions of Lipschitz BSDEs in the case of bounded coefficients. In the case of convex constraints and under some exponential integrability assumptions on the coefficients, some complementary properties are provided. These results can be generalized to the case of several default times or a Poisson process.