How Close Is Bayes GZ to the True Optimum?

Abstract

Sukhov (2026c) derives the Bayesian Grossman–Zhou rule, f_BGZ(d,n) = κ̄(n)d/b, as the tractable stationary benchmark for Kelly betting under a drawdown barrier and parameter uncertainty, and leaves its distance from the true optimum as an open numerical question bounded heuristically at O(r/μ) ≈ 13%. This paper closes that question. We first identify and correct a missing Itô-curvature term in the governing HJB equation; the corrected first-order condition shows analytically that the linear Bayes GZ rule is the exact leading-order optimum near the drawdown barrier, but that the optimality gap away from the barrier is governed entirely by the curvature of the value function at the running maximum, V_dd(b). We then show, by direct CRN Monte-Carlo optimisation over stationary policies (bootstrap CIs, N = 15–60k paths) cross-validated against a finite-difference HJB solve, that this gap decomposes into two empirically separable channels: an amplitude channel (how aggressively to delever) and a shape channel (linear vs. concave cushion). Which channel dominates is regime-dependent: under known-edge conditions the shape channel is active and significant (ΔU = +0.0064, 95% CI [+0.0056, +0.0072]); under realistic parameter uncertainty the entire gap is amplitude (+0.08 to +0.25 across estimation horizons) and the shape channel contributes ≈ 0 — Bayes GZ’s linear cushion is correct, its shrinkage is too timid. A single fixed replacement rule, calibrated only on GBM, dominates Bayes GZ across five return environments (GBM, Markov regime-switching, Student-t, jump-diffusion, GARCH), including the two environments where Sukhov (2026c) reported Bayes GZ losing to plug-in Kelly. At monthly rebalancing, the linear Bayes GZ rule is outright ruinous (U = −0.25) while the concave replacement survives (U = +0.06). We map the parameter region where the gap is economically material: it grows monotonically with the perceived edge κ̄ and with barrier tightness, from under 10% at modest leverage to over 100% at high leverage under a tight 10% drawdown mandate.

Keywords

• Kelly criterion
• drawdown constraint
• parameter uncertainty
• HJB equation
• stochastic control

This paper is forthcoming. The abstract above is the current draft; the SSRN record and citation details will appear here when it is posted.

Research area

Drawdown-Constrained Capital Allocation