Risk-Constrained Kelly for Mutually Exclusive Outcomes: CRRA Support Invariance and Logarithmic One-Dimensional Calibration

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The paper is highly relevant to Kelly-based portfolio construction and risk-constrained betting strategies. The support invariance result (risk constraint does not change which assets are held, only their weights) is practically important for portfolio managers. The structured O(n log n) solver is directly implementable. However, the mutually exclusive outcome assumption limits direct applicability to continuous financial markets. The theoretical framework is more relevant to discrete betting/gambling contexts and structured product allocation than to typical equity portfolio optimization.

The structured solver (Corollary 5.4) is algorithmically straightforward: sort likelihood ratios, find prefix index, evaluate calibration functional, solve one scalar equation via bisection/Newton, then reconstruct from closed-form inner solutions. The main complexity lies in understanding the theoretical framework and correctly implementing the inner one-dimensional solves. For the logarithmic case, the inner equations have explicit solutions (e.g., quadratic for lambda=1). The overround regime assumption simplifies the problem considerably.

The paper is self-contained with complete proofs, explicit formulas, and a numerical example (Section 7) with all parameters specified. The structured solver algorithm (Corollary 5.4) is clearly stated with O(n log n) complexity. However, no code repository is provided, and the numerical example is small (3 outcomes). All mathematical derivations are fully detailed.