Logarithmic regret in the ergodic Avellaneda-Stoikov market making model

Jialun Cao, David \v{S}i\v{s}ka, Lukasz Szpruch and Tanut Treetanthiploet

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Abstract: We analyse the regret arising from learning the price sensitivity parameter $\kappa$ of liquidity takers in the ergodic version of the Avellaneda-Stoikov market making model. We show that a learning algorithm based on a maximum-likelihood estimator for the parameter achieves the regret upper bound of order $\ln^2 T$ in expectation. To obtain the result we need two key ingredients. The first is the twice differentiability of the ergodic constant under the misspecified parameter in the Hamilton-Jacobi-Bellman (HJB) equation with respect to $\kappa$, which leads to a second--order performance gap. The second is the learning rate of the regularised maximum-likelihood estimator which is obtained from concentration inequalities for Bernoulli signals. Numerical experiments confirm the convergence and the robustness of the proposed algorithm.

Date: 2024-09, Revised 2025-07
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