Maximisation de l'utilité exponentielle dans un marché incomplet avec défaut

Résumé

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 stock. We first consider the case of strategies valued in a compact set. Using a verification theorem, we prove that the value function associated with the optimization problem can be characterized as the solution of a Lipschitz BSDE (backward stochastic differential equation). Then, we consider the case of non constrained strategies. Using dynamic programming techniques, we prove that the value function is the maximal solution of a BSDE. Moreover, the value function is the limit of a sequence of processes which are the solutions of Lipschitz BSDEs. These properties can be generalized to the case of several default times or a Poisson process.

Codes JEL