Efficient Consumption Set Under Recursive Utility and Unknown Beliefs

In a context of complete financial markets where asset prices follow Ito's processes, we characterize the set of consumption processes which are optimal for a given stochastic differential utility (e.g. Duffie and Epstein (1992)) when beliefs are unknown. Necessary and sufficient conditions for the efficiency of a consumption process, consists of the existence of a solution to a quadratic backward stochastic differential equation and a martingale condition. We study the efficiency condition in the case of a class of homothetic stochastic differential utilities and derive some results for those particular cases. In a Markovian context, this efficiency condition becomes a partial differential equation.