Papers
WP·2026·003 · June 2026 · Risk · Bayesian

The Bayesian Grossman–Zhou Rule

Drawdown-Constrained Kelly Betting with Parameter Uncertainty

By · Market Microstructure Research Lab

Abstract

The Kelly criterion assumes that expected returns are known. Real investors face the opposite problem: leverage must be chosen before the edge is known with confidence. At the same time, most trading mandates impose hard drawdown limits that can terminate a strategy after a relatively small loss from peak. These two frictions — learning and survival — are usually studied separately. This paper combines them in a single continuous-time framework. We formulate a two-dimensional Hamilton–Jacobi–Bellman equation in which the investor simultaneously manages drawdown risk and parameter uncertainty. The state variables are the log-distance to the drawdown barrier and an endogenous measure of information quality generated through Bayesian learning. The analysis reveals a structural result. Under the undiscounted growth criterion, the problem admits a tractable benchmark policy, the Bayesian Grossman–Zhou (Bayes GZ) rule, which combines Bayesian shrinkage with drawdown-sensitive leverage. Under positive discounting, however, no separable linear policy can satisfy the HJB: the value of learning creates nonlinear coupling between risk-taking and information acquisition. The resulting policy exhibits “double prudence”: leverage increases only when both the drawdown state and estimation quality improve. We further establish exact O(1/n) convergence to the oracle allocation as information accumulates. Monte Carlo experiments based on 50,000 simulated paths show that under cold-start conditions (one year of data), Bayes GZ improves survival probability by 11 percentage points and increases penalised utility relative to plug-in Kelly sizing. More broadly, the results suggest that leverage is not merely a response to information — it is also a mechanism for generating it.

Keywords

  • Kelly criterion
  • Bayesian learning
  • Drawdown constraint
  • HJB equation
  • Grossman–Zhou
  • Parameter uncertainty

Cite this paper

Sukhov, S. (2026). The Bayesian Grossman–Zhou Rule: Drawdown-Constrained Kelly Betting with Parameter Uncertainty. MMRL Working Paper WP-2026-003. SSRN. https://doi.org/10.2139/ssrn.6942459

@techreport{sukhov2026bayesian,
  author      = {Sukhov, Sergei},
  title       = {The Bayesian Grossman--Zhou Rule: Drawdown-Constrained Kelly Betting with Parameter Uncertainty},
  institution = {Market Microstructure Research Lab},
  type        = {MMRL Working Paper},
  number      = {WP-2026-003},
  year        = {2026},
  month       = jul,
  doi         = {10.2139/ssrn.6942459},
  url         = {https://papers.ssrn.com/sol3/papers.cfm?abstract_id=6942459},
}

Research area

Drawdown-Constrained Capital Allocation