Paper: arXiv 2409.19259

Authors: Jiaqin Wei, Jianming Xia, Qian Zhao

Abstract

We investigate the portfolio selection problem for an agent with rank-dependent utility in an incomplete financial market. For a constant-coefficient market and CRRA utilities, we characterize the deterministic strict equilibrium strategies. In the case of time-invariant probability weighting function, we provide a comprehensive characterization of the deterministic strict equilibrium strategy. The unique non-zero equilibrium, if exists, can be determined by solving an autonomous ODE. In the case of time-variant probability weighting functions, we observe that there may be infinitely many non-zero deterministic strict equilibrium strategies, which are derived from the positive solutions to a nonlinear singular ODE. By specifying the maximal solution to the singular ODE, we are able to identify all the positive solutions. In addition, we address the issue of selecting an optimal strategy from the numerous equilibrium strategies available.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper offers a rigorous theoretical analysis of time-consistent portfolio selection under rank-dependent utility in incomplete markets, solving complex non-linear singular ODEs. However, it is entirely theoretical with no empirical backtests or market data analysis.

Research Flowchart

  flowchart TD
    A[Research Goal: Time-Consistent Portfolio Selection under RDU in Incomplete Markets] --> B[Inputs: CRRA Utilities, Constant Coefficients & Probability Weighting Functions]
    B --> C[Analyze Time-Consistency & Equilibrium Strategies]
    
    C --> D{Weighting Function Type?}
    
    D -- Time-Invariant --> E[Solve Autonomous ODE]
    D -- Time-Variant --> F[Analyze Nonlinear Singular ODE & Maximal Solution]
    
    E --> G[Outcomes: Unique Non-Zero Equilibrium Strategy]
    F --> H[Outcomes: Infinitely Many Equilibria & Selection of Optimal Strategy]

Cited References (60)

  1. Forward rank-dependent performance criteria: Time-consistent investment under probability distortion (2021)
  2. The golden rule when preferences are time inconsistent (2010)
  3. Being serious about non-commitment: subgame perfect equilibrium in continuous time (2006) — link
  4. Optimal equilibria for time-inconsistent stopping problems in continuous time (2020)
  5. The Optimal Equilibrium for Time-Inconsistent Stopping Problems—The Discrete-Time Case (2019)
  6. Optimal Equilibria for Multidimensional Time-Inconsistent Stopping Problems (2021)
  7. The Dual Theory of Choice under Risk (1987)
  8. A Class of Distortion Operators for Pricing Financial and Insurance Risks (2000)
and 52 more
  1. Ordinary Differential Equations (1998)
  2. A synthesis of risk measures for capital adequacy (1999)
  3. Optimum consumption and portfolio rules in a continuous-time model (1971)
  4. Risk aversion in the theory of expected utility with rank dependent probabilities (1987)
  5. Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case (1969)
  6. On time-inconsistent stochastic control in continuous time (2017)
  7. Investment and consumption without commitment (2008)
  8. Time-Consistent Portfolio Management (2012)
  9. Time-inconsistent optimal control problems and the equilibrium HJB equation (2012)
  10. LINEAR-QUADRATIC OPTIMAL CONTROL PROBLEMS FOR MEAN-FIELD STOCHASTIC DIFFERENTIAL EQUATIONS — TIME-CONSISTENT SOLUTIONS (2017)
  11. Time-Consistent Mean-Variance Asset-Liability Management with Random Coefficients (2017)
  12. Time-Inconsistent Stochastic Linear–Quadratic Control: Characterization and Uniqueness of Equilibrium (2017)
  13. Time-inconsistent stochastic linear-quadratic control (2012)
  14. Dynamic mean-variance asset allocation (2010)
  15. Mean–Variance Portfolio Optimization with State-Dependent Risk Aversion (2014)
  16. Myopia and inconsistency in dynamic utility maximization (1955)
  17. Arrow–Debreu equilibria for rank-dependent utilities with heterogeneous probability weighting (2019)
  18. A theory of anticipated utility (1982)
  19. Le Comportement de l’Homme Rationnel devant le Risque: Critique des Postulats et Axiomes de l’Ecole Americaine (1953)
  20. Risk, Ambiguity, and the Savage Axioms (1961)
  21. Sustainability: Game human nature (2016) — link
  22. On the equilibrium strategies for time-inconsistent problems in continuous time (2021)
  23. Strong and Weak Equilibria for Time-Inconsistent Stochastic Control in Continuous Time (2021)
  24. Consistent investment of sophisticated rank-dependent utility agents in continuous time (2021)
  25. From T anaka’s formula to I to’s formula: the fundamental theorem of stochastic calculus (1997)
  26. Optimal Control Models in Finance: A New Computational Approach (2005)
  27. Numerical Methods for Optimal Control Problems with State Constraints (1999)
  28. Optimal Control from Theory to Computer Programs (2003)
  29. HOPE, FEAR, AND ASPIRATIONS (2016) — link
  30. On the Neyman-Pearson problem for law-invariant risk measures and robust utility functionals (2004)
  31. Continuous Martingales and Brownian Motion (1999)
  32. Optimization by Vector Space Methods (1969)
  33. Probability and Stochastics (2011)
  34. Measure Theory (1994)
  35. A Note on the quantile formulation (2016)
  36. Arrow- D ebreu equilibria for rank-dependent utilities (2016)
  37. Necessary and sufficient conditions in the problem of optimal investment in incomplete markets (2003)
  38. The asymptotic elasticity of utility functions and optimal investment in incomplete markets (1999)
  39. Prospect Theory: For Risk and Ambiguity (2010)
  40. A new characterization of comonotonicity and its application in behavioral finance (2014)
  41. Law invariant concave utility functions and optimization problems with monotonicity and comonotonicity constraints (2006)
  42. OPTIMAL DEMAND FOR CONTINGENT CLAIMS WHEN AGENTS HAVE LAW INVARIANT UTILITIES (2011)
  43. Portfolio Choice via Quantiles (2011)
  44. Behavioral portfolio selection in continuous time (2008)
  45. Stochastic Finance: an Introduction in Discrete Time (2016)
  46. Expected Utility Maximization with Stochastic Dominance Constraints in Complete Markets (2021)
  47. Convexity and Optimization in Banach Spaces (2012)
  48. Reapted Games (2015)
  49. Convex Optimization in Normed Spaces Theory, Methods and Examples (2015)
  50. An Introduction to Optimal Control Problems in Life Sciences and Economics: From Mathematical Models to Numerical Simulation with MATLAB (2011)
  51. A Survey of Recent Results for Initial and Boundary Value Problems Singular in the Dependent Variable (2004)
  52. Dynamic portfolio selection under generalized disappointment aversion (2024) — link

0 of 60 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.