  This paper considers a class of stochastic control problems with implicitly
defined objective functions, which are the sources of time-inconsistency. We
study the closed-loop equilibrium solutions in a general controlled diffusion
framework. First, we provide a sufficient and necessary condition for a
strategy to be an equilibrium. Then, we apply the result to discuss two
problems of dynamic portfolio selection for a class of betweenness preferences,
allowing for closed convex constraints on portfolio weights and borrowing cost,
respectively. The equilibrium portfolio strategies are explicitly characterized
in terms of the solutions of some first-order ordinary differential equations
for the case of deterministic market coefficients.
