Optimal Stochastic Control Problem Under Model Uncertainty With Nonentropy Penalty

In this paper, a stochastic control problem under model uncertainty with general penalty term is studied. Two types of penalties are considered. The first one is of type [Formula: see text]-divergence penalty treated in the general framework of a continuous filtration. The second one called consistent time penalty is studied in the context of a Brownian filtration. In the case of consistent time penalty, we characterize the value process of our stochastic control problem as the unique solution of a class of quadratic backward stochastic differential equation with unbounded terminal condition.