  This paper addresses the portfolio selection problem for nonlinear
law-dependent preferences in continuous time, which inherently exhibit time
inconsistency. Employing the method of stochastic maximum principle, we
establish verification theorems for equilibrium strategies, accommodating both
random market coefficients and incomplete markets. We derive the first-order
condition (FOC) for the equilibrium strategies, using a notion of functional
derivatives with respect to probability distributions. Then, with the help of
the FOC we obtain the equilibrium strategies in closed form for two classes of
implicitly defined preferences: CRRA and CARA betweenness preferences, with
deterministic market coefficients. Finally, to show applications of our
theoretical results to problems with random market coefficients, we examine the
weighted utility. We reveal that the equilibrium strategy can be described by a
coupled system of Quadratic Backward Stochastic Differential Equations
(QBSDEs). The well-posedness of this system is generally open but is
established under the special structures of our problem.
