Paper: arXiv 2604.11577

Authors: Christopher D. Long

Abstract

We study the finite mutually exclusive outcome version of risk-constrained Kelly optimization with explicit state prices. The market has outcome probabilities $p_i>0$, state prices $q_i>0$, terminal wealths $W_i=c+x_i/q_i$, and a drawdown-surrogate constraint [ \sum_{i=1}^n p_i W_i^{-λ}\le 1,\qquad λ>0. ] For constant relative risk aversion utility, we work primarily in the standard overround regime $\sum_i q_i>1$, where every optimizer is necessarily non-full-support. Under the usual unique likelihood-ratio prefix hypothesis for the unconstrained problem, we prove that the constrained optimizer has exactly the same active set. Thus, in the regime where the prefix theorem is meaningful, the risk constraint deforms the funded wealth profile but does not change the active set. The support is therefore invariant across both the CRRA parameter and the drawdown-surrogate parameter. We then isolate the logarithmic case $γ=1$. Once the common active prefix is known, the constrained problem reduces to a one-dimensional outer calibration together with independent one-dimensional inner equations on the active states. In this case we prove existence, uniqueness, and monotonicity for the inner solves, derive a complete calibration theorem, and record the resulting structured algorithm. We treat the fair and subfair regimes only as boundary cases: full-support phenomena can occur there, so the overround prefix theory no longer yields a parallel exact description of comparable sharpness. A numerical example illustrates how the risk constraint alters the funded wealth profile while leaving support unchanged.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous theoretical extension of Kelly criterion optimization under drawdown-surrogate constraints, proving support invariance for CRRA utilities. While mathematically strong with clear proofs and algorithmic reductions, it lacks empirical validation, real-world backtesting, or empirical market data.

Research Flowchart

  flowchart TD
    A[Research Goal: Solve Risk-Constrained Kelly<br/>Optimization with Drawdown Constraint] --> B[Input Data: Probabilities p_i, State Prices q_i,<br/>Drawdown Parameter λ, Risk Aversion γ]
    B --> C[CRRA Analysis in Overround Regime<br/>Sum q_i > 1]
    C --> D[Prove Active Set Invariance:<br/>Support is invariant to γ and λ]
    D --> E[Isolate Logarithmic Case γ=1]
    E --> F[1D Outer Calibration & Independent<br/>1D Inner Solves on Active States]
    F --> G[Key Outcome: Efficient Algorithm & Proof that<br/>Risk Constraint Alters Wealth Profile, Not Support]