Paper: arXiv 2311.06745

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.

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: 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.

Research Flowchart

  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"]

Cited References (57)

  1. Time-Inconsistent Stochastic Linear-Quadratic Control: Characterization and Uniqueness of Equilibrium (2017) — link
  2. A theory of Markovian time-inconsistent stochastic control in discrete time (2014)
  3. Investment and consumption without commitment (2008)
  4. Optimal Dynamic Portfolio Selection: Multiperiod Mean-Variance Formulation (2000)
  5. Continuous-Time Mean-Variance Portfolio Selection: A Stochastic LQ Framework (2000)
  6. Time-inconsistent stochastic linear-quadratic control (2012)
  7. Time-Consistent Portfolio Management (2012)
  8. The golden rule when preferences are time inconsistent (2010)
and 49 more
  1. Optimum consumption and portfolio rules in a continuous-time model (1971)
  2. Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case (1969)
  3. Portfolio Selection (1952)
  4. Myopia and Inconsistency in Dynamic Utility Maximization (1955)
  5. Existence and Uniqueness for Non-Markovian Triangular Quadratic BSDEs (2022)
  6. Multi-dimensional backward stochastic differential equations of diagonally quadratic generators: The general result (2023)
  7. Multi-dimensional backward stochastic differential equations of diagonally quadratic generators (2016)
  8. On time-inconsistent stochastic control in continuous time (2017)
  9. Generalized Expected Utility Theory: The Rank-Dependent Model (1993)
  10. A theory of anticipated utility (1982)
  11. Backward Stochastic Differential Equations: From Linear to Fully Nonlinear Theory (2017)
  12. Optimal Portfolio Selection under Disappointment Averse Utility (2010) — link
  13. Being serious about non-commitment: subgame perfect equilibrium in continuous time (2006) — link
  14. Multidimensional quadratic BSDEs with separated generators (2017) — link
  15. Probabilistic Theory of Mean Field Games with Applications I: Mean Field FBSDEs, Control, and Games (2018)
  16. A rank-dependent generalization of zero utility principle (2003)
  17. Equilibrium investment with random risk aversion (2023)
  18. Le Comportement de l’Homme Rationnel devant le Risque: Critique des Postulats et Axiomes de l’Ecole Americaine (1953)
  19. Axiomatic utility theories with the betweenness property (1989)
  20. An Operational Measure of Riskiness (2009)
  21. An Economic Index of Riskiness (2008)
  22. A Generalization of the Quasilinear Mean with Applications to the Measurement of Income Inequality and Decision Theory Resolving the Allais Paradox (1983)
  23. Asset Pricing and Portfolio Choice Theory (2017)
  24. Learning equilibrium mean-variance strategy (2023)
  25. A dynamic mean-variance analysis for log returns (2021)
  26. An axiomatic characterization of preferences under uncertainty: Weakening the independence axiom (1986)
  27. Splitting multidimensional BSDEs and finding local equilibria (2014)
  28. A system of quadratic BSDEs arising in a price impact model (2016) — link
  29. Solvability of backward stochastic differential equations with quadratic growth (2008)
  30. A Theory of Disappointment Aversion (1991)
  31. Generalized disappointment aversion, long-run volatility risk, and asset prices (2011)
  32. Generalized Disappointment Aversion and Asset Prices (2010)
  33. Me, Myself and I: A general theory of non-Markovian time-inconsistent stochastic control for sophisticated agents (2023)
  34. Extended backward stochastic volterra integral equations and their applications to time-inconsistent stochastic recursive control problems (2021) — link
  35. Equilibrium asset pricing with transaction costs (2021) — link
  36. Utility maximization in incomplete markets (2005)
  37. Continuous Exponential Martingales and BMO (1994)
  38. Brownian motion and stochastic calculus (1998)
  39. Mean field portfolio games (2023)
  40. Completeness of security markets and solvability of linear backward stochastic differential equations (2006)
  41. Time-inconsistent consumption-investment problems in incomplete markets under general discount functions (2021) — link
  42. A General Chain Rule for Derivatives and the Change of Variables Formula for the Lebesgue Integral (1969)
  43. Stochastic Controls (1999)
  44. Multidimensional Diffusion Processes (1997)
  45. Me, myself and I: a general theory of non-Markovian time-inconsistent stochastic control for sophisticated agents (2020) — link
  46. Equilibrium master equations for time-inconsistent problems with distribution dependent rewards (2021) — link
  47. Mean-variance portfolio optimization with state-dependent risk aversion (2014)
  48. Dynamic Mean-Variance Asset Allocation (2010)
  49. Consistent investment of sophisticated rank-dependent utility agents in continuous time (2021) — link

0 of 57 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.