Dynamic portfolio selection for nonlinear law-dependent preferences

ArXiv ID: 2311.06745 “View on arXiv”

Authors: Unknown

Abstract

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.

Keywords: portfolio selection, time inconsistency, stochastic maximum principle, backward stochastic differential equations (BSDE), equilibrium strategies, Portfolio Management

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: The paper is heavily theoretical, employing advanced stochastic calculus, maximum principle, and BMO martingale theory to derive general verification theorems for time-inconsistent preferences, with no data or backtesting mentioned. The examples provided are analytical derivations (closed-form solutions, QBSDEs) rather than empirical validation, placing it squarely in the theoretical research quadrant.

  flowchart TD
    A["Research Goal:<br>Dynamic portfolio selection for<br>nonlinear law-dependent preferences"] --> B["Key Methodology:<br>Stochastic Maximum Principle &<br>Functional Derivatives w.r.t. Probabilities"]
    B --> C["Establish Verification Theorems<br>& First-Order Condition (FOC)"]
    C --> D{"Computational Process:<br>Solve for Equilibrium Strategies"}
    D --> E["Case 1: CRRA/CARA Preferences<br>(Deterministic Coefficients)<br>Outcome: Closed-form solutions"]
    D --> F["Case 2: Weighted Utility<br>(Random Coefficients)<br>Outcome: Coupled QBSDE System"]
    E --> G["Key Outcomes"]
    F --> G
    G --> H["General well-posedness of QBSDEs open,<br>but established for problem structure"]