  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 f-divergence penalty treated in the general framework of a
continuous filtration. The second one called consistent time penalty 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.
