Dynamic Portfolio Selection for Nonlinear Law-Dependent Preferences

This paper addresses the portfolio selection problem for nonlinear law-dependent preferences in continuous time, which inherently exhibit time inconsistency. Employing the method of the 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: constant relative risk aversion and constant absolute risk aversion 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. The well-posedness of this system is generally open but is established under the special structures of our problem. Funding: This work was supported by the National Key Research and Development Program of China [Grant 2020YFA0712700] and the National Natural Science Foundation of China [Grants 11871036, 12071146, 12271290, 12431017, and 12471447]. J. Xia also acknowledges support from the Key Laboratory of Random Complex Structures and Data Science, National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences. Supplemental Material: The online companion is available at https://doi.org/10.1287/moor.2023.0345 .