  We study portfolio selection in a complete continuous-time market where the
preference is dictated by the rank-dependent utility. As such a model is
inherently time inconsistent due to the underlying probability weighting, we
study the investment behavior of sophisticated consistent planners who seek
(subgame perfect) intra-personal equilibrium strategies. We provide sufficient
conditions under which an equilibrium strategy is a replicating portfolio of a
final wealth. We derive this final wealth profile explicitly, which turns out
to be in the same form as in the classical Merton model with the market price
of risk process properly scaled by a deterministic function in time. We present
this scaling function explicitly through the solution to a highly nonlinear and
singular ordinary differential equation, whose existence of solutions is
established. Finally, we give a necessary and sufficient condition for the
scaling function to be smaller than 1 corresponding to an effective reduction
in risk premium due to probability weighting.
