Paper: arXiv 2505.20504

Authors: Peter Holm Nielsen

Abstract

We propose martingale consumption as a natural, desirable consumption pattern for any given (proportional) investment strategy. The idea is to always adjust current consumption so as to achieve level expected future consumption under the arbitrarily chosen investment strategy. This approach avoids the formulation of an optimization objective based on preferences towards risk, intertemporal consumption, habit formation etc. We identify general explicit solutions in deterministic-coefficient models. In the general case with random coefficients we establish uniqueness, but the question of existence of a solution is unsettled. With the interest rate as the only random factor we derive a PDE for the wealth-to-consumption factor as a function of the state variables, which, however, is non-linear and without known closed-form solutions. We briefly consider the discrete-time case and obtain similar results. Throughout, we compare with well-known optimal strategies for classical CRRA investors with time-additive utility of consumption and find that under suitable time preferences they may in certain cases achieve martingale consumption simultaneously.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 1.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper introduces an innovative theoretical framework (Martingale Consumption) that bypasses traditional utility function optimization, utilizing solid continuous-time stochastic calculus and non-linear PDEs. However, it is purely theoretical with no empirical data, backtesting, or empirical validation included in the provided text.

Research Flowchart

  flowchart TD
    A[Research Goal: Define Martingale Consumption for any Investment Strategy] --> B[Inputs: Proportional Investment Strategy & Asset Price Dynamics]
    B --> C[Methodology: Adjust Consumption for Level Expected Future Consumption]
    C --> D[Model Analysis]
    
    D --> E[Deterministic Coefficients: Obtain Explicit Solutions]
    D --> F[Stochastic Interest Rates: Derive Non-Linear PDE for Wealth-to-Consumption]
    
    E & F --> G[Outcomes: Prove Uniqueness, Identify Open Existence Questions & Show Alignment with Specific CRRA Utility Models]