Before revision:  We provide financial interpretations of the optimal measure transformation problems in terms of the  maximization of returns subject to a relative entropy penalty term. In general the solution of the optimal measure transformation problem is characterized by the solution of certain decoupled nonlinear forward-backward stochastic differential equations (FBSDEs) .
After revision:  We provide financial interpretations of the optimal measure transformation problems in terms of the maximization of returns subject to a penalty for removing financial risk as expressed through the aggregate relative entropy .