Time-consistent portfolio selection with strictly monotone mean-variance preference

Paper: arXiv 2502.11052

Abstract

This paper is devoted to time-consistent control problems of portfolio selection with strictly monotone mean-variance preferences. These preferences are variational modifications of the conventional mean-variance preferences, and remain time-inconsistent as in mean-variance optimization problems. To tackle the time-inconsistency, we study the Nash equilibrium controls of both the open-loop type and the closed-loop type, and characterize them within a random parameter setting. The problem is reduced to solving a flow of forward-backward stochastic differential equations for open-loop equilibria, and to solving extended Hamilton-Jacobi-Bellman equations for closed-loop equilibria. In particular, we derive semi-closed-form solutions for these two types of equilibria under a deterministic parameter setting. Both solutions are represented by the same function, which is independent of wealth state and random path. This function can be expressed as the conventional time-consistent mean-variance portfolio strategy multiplied by a factor greater than one. Furthermore, we find that the state-independent closed-loop Nash equilibrium control is a strong equilibrium strategy in a constant parameter setting only when the interest rate is sufficiently large.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper relies heavily on advanced stochastic control theory, forward-backward SDEs, and extended HJB equations, indicating high mathematical complexity. However, the analysis is purely theoretical with no empirical data, backtesting, or implementation details, resulting in low empirical rigor.

Research Flowchart

  flowchart TD
  A["Research Goal<br>Find time-consistent<br>portfolio selection<br>for strict monotone<br>mean-variance preference"] --> B{"Methodology<br>Nash Equilibrium Analysis"}

  B --> C["Type 1: Open-Loop<br>Solve Forward-Backward<br>Stochastic Differential Equations"]
  B --> D["Type 2: Closed-Loop<br>Solve Extended<br>HJB Equations"]

  C --> E["Compute Equilibria"]
  D --> E

  E --> F["Key Findings/Outcomes<br>1. Semi-closed solutions<br>2. Strategy = Conventional MV<br>Strategy × Factor > 1<br>3. Wealth-Independent<br>4. Strong equilibrium<br>requires high interest rate"]

Cited References (52)

1. Dynamic asset allocation under mean-reverting returns, stochastic interest rates, and inflation uncertainty: A re popular recommendations consistent with rational behavior? (2004) — link
2. Interest Rate Statistics: D aily Treasury Par Yield Curve Rates (2026) — link
3. Characterizations of closed-loop equilibrium solutions for dynamic mean-variance optimization problems (2017) — link
4. Dynamic Mean-Variance Asset Allocation (2010) — link
5. Convex Analysis and Monotone Operator Theory in H ilbert Spaces (2017) — link
6. L$^P$ solutions of backward stochastic differential equations (2003) — link
7. Dynamic asset allocation under inflation (2002) — link
8. On time-inconsistent stochastic control in continuous time (2017) — link

4 of 52 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.