Maximization of Recursive Utilities: A Dynamic Maximum Principle Approach

We study a maximization problem from terminal wealth and consumption for a class of robust utility functions introduced in Bordigoni, Matoussi, and Schweizer [A stochastic control approach to a robust utility maximization problem, in Stochastic Analysis and Applications, Abel Symp. 2, F. E. Benth, G. Di Nunno, T. Lindstrom, B. Oksendal, and T. Zhang, eds., Springer, Berlin, 2007, pp. 125-151]. Our method is based on backward stochastic differential equation theory techniques. We prove a dynamic maximum principle for the optimal control. We study the existence and the uniqueness of the consumption-investment strategy which is characterized as the unique solution of a forward-backward system.