Consumption-Investment Problem in Rank-Based Models

Paper: arXiv 2510.20763

Authors: David Itkin

Abstract

We study a consumption-investment problem in a multi-asset market where the returns follow a generic rank-based model. Our main result derives an HJB equation with Neumann boundary conditions for the value function and proves a corresponding verification theorem. The control problem is nonstandard due to the discontinuous nature of the coefficients in rank-based models, requiring a bespoke approach of independent mathematical interest. The special case of first-order models, prescribing constant drift and diffusion coefficients for the ranked returns, admits explicit solutions when the investor is either (a) unconstrained, (b) abides by open market constraints or (c) is fully invested in the market. The explicit optimal strategies in all cases are related to the celebrated solution to Merton’s problem, despite the intractability of constraint (b) in that setting.

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 provides a highly rigorous theoretical treatment of the consumption-investment problem under rank-based models, deriving novel explicit solutions for HJB equations with Neumann boundary conditions. However, it is purely mathematical without empirical backtesting or real market data validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Solve Consumption-Investment Problem in Rank-Based Models] --> B[Input: Multi-Asset Market with Generic Rank-Based Returns]
    B --> C[Formulate Stochastic Control Problem with Discontinuous Coefficients]
    C --> D[Derive HJB Equation with Neumann Boundary Conditions & Verification Theorem]
    D --> E[Analyze First-Order Model Case: Constant Drift & Diffusion]
    E --> F[Evaluate Constraints: Unconstrained, Open Market, & Fully Invested]
    F --> G[Key Outcome: Explicit Optimal Strategies Related to Merton's Problem]