Paper: arXiv 2602.18157

Authors: Oumar Mbodji

Abstract

We study the Merton portfolio management problem within a complete market, non constant time discount rate and general utility framework. The non constant discount rate introduces time inconsistency which can be solved by introducing sub game perfect strategies. Under some asymptotic assumptions on the utility function, we show that the subgame perfect strategy is the same as the optimal strategy, provided the discount rate is replaced by the utility weighted discount rate $ρ(t,x)$ that depends on the time $t$ and wealth level $x$. A fixed point iteration is used to find $ρ$. The consumption to wealth ratio and the investment to wealth ratio are given in feedback form as functions of the value function.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical derivation of subgame perfect strategies for time-inconsistent Merton problems under general utility functions using extended HJB equations. However, it is purely theoretical with zero empirical backtesting or real-world data validation.

Research Flowchart

  flowchart TD
    A["<b>Research Goal:</b> Solve time-inconsistent Merton problem for general utility & non-constant discount rate"] --> B["<b>Inputs:</b> Complete market model, non-constant discount rate, general utility function"]
    
    B --> C["<b>Methodology:</b> Formulate game-theoretic framework for subgame perfect equilibrium"]
    
    C --> D["<b>Computation:</b> Derive utility-weighted discount rate ρ(t,x) via fixed-point iteration"]
    
    D --> E["<b>Key Outcome:</b> Subgame perfect strategy equals optimal strategy using ρ(t,x)"]
    
    E --> F["<b>Final Output:</b> Consumption-to-wealth & investment-to-wealth ratios in feedback form"]