Paper: arXiv 2610.10046

Authors: Gechun Liang, Moris S. Strub, Yuwei Wang, Zhaojun Yang

Abstract

We develop a framework for an investor who trades until she either reaches a financial goal or an exogenous deadline arrives. Analogous to utility functions over wealth, we measure satisfaction with the timing of reaching a goal by a discount function. For a continuous-time market where a stochastic factor drives the dynamics of stock prices and the financial goals, the investor maximizes the expected discount at the goal reaching time and the expected utility of the funding ratio if the goal remains unreached by the deadline. This setup leads to a new class of stochastic control problems. We establish Bellman’s principle of optimality and characterize the value function as a viscosity solution to the associated Hamilton-Jacobi-Bellman equation. When the deadline is infinite and the goal is constant, the HJB equation reduces to a form related to the backward heat equation, for which we provide a complete characterization of smooth solutions. When analytical solutions are unavailable, we develop an approach based on Howard’s algorithm to obtain numerical solutions. Our analysis shows that optimal investment policies for goal reaching problems can be decreasing in the drift of risky assets and need not converge to full risk-free investment even as volatility diverges to infinity.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a highly mathematical and novel stochastic control framework for goal-based investing. While it offers significant theoretical contributions, the empirical rigor is limited, focusing on analytical and numerical solutions rather than extensive backtesting or real-world data validation. The paper introduces a new class of stochastic control problems and provides a thorough theoretical treatment.

Research Flowchart

  flowchart TD
    A[Research Goal: Optimal Investment for Financial Goal] --> B{Stochastic Control Framework};
    B --> C{Methodology: HJB Equation, Viscosity Solutions, Howard's Algorithm};
    C --> D[Inputs: Continuous-Time Market (Stochastic Factor), Discount Function, Utility Function];
    D --> E{Computational Process: Analytical Solutions (Backward Heat Eq. form), Numerical Solutions (Howard's Algo.)};
    E --> F[Key Findings: Optimal policies decreasing in risky asset drift, no full risk-free investment at high volatility];

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