Paper: arXiv 2305.07166

Authors: Mengge Li, Shuaijie Qian, Chao Zhou

Abstract

The classical mean-variance portfolio selection problem induces time-inconsistent (precommited) strategies (see Zhou and Li (2000)). To overcome this time-inconsistency, Basak and Chabakauri (2010) introduce the game theoretical approach and look for (sub-game perfect Nash) equilibrium strategies, which is solved from the corresponding partial differential equations (PDE) system. In their model, the investor perfectly knows the drift and volatility of the assets. However, in reality investors only have an estimate on them, e.g, a 95% confidence interval. In this case, some literature (e.g., Pham, Wei and Zhou (2022)) derives the optimal precommited strategy under the worst parameters, which is the robust control. The relation between the equilibrium strategy and the PDE system has not been justified when incorporating robust control. In this paper, we consider a general dynamic mean-variance framework and propose a novel definition of the robust equilibrium strategy. Under our definition, a classical solution to the corresponding PDE system implies a robust equilibrium strategy. We then explicitly solve for some special examples.

Complexity vs Empirical Score

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

Why this score: The paper tackles a mathematically sophisticated problem by merging time-inconsistent dynamic programming, game-theoretic Nash equilibrium, robust control, and systems of PDEs. However, it is entirely theoretical with no empirical backtesting, datasets, or code provided, leaving it in the ‘Lab Rats’ quadrant.

Research Flowchart

  flowchart TD
    A[Research Goal: Define Robust Equilibrium Strategy for Time-Inconsistent Dynamic Mean-Variance Selection] --> B[Inputs: Asset Parameters with Uncertain Drift & Volatility]
    B --> C[Methodology: Combine Robust Control Worst-Case with Game Theoretical Approach]
    C --> D[Mathematical Formulation: Derive Corresponding Non-linear PDE System]
    D --> E[Computation: Solve PDE System for Classical Solutions & Explicit Examples]
    E --> F[Outcomes: Proved Classical PDE Solution Implies Robust Equilibrium Strategy & Derived Explicit Solutions]