Logarithmic Regret in the Ergodic Avellaneda–Stoikov Market Making Model

Abstract. We analyze the regret arising from learning the price sensitivity parameter [Formula: see text] 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 [Formula: see text] 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 equation with respect to [Formula: see text], which leads to a second-order performance gap. The second is the learning rate of the regularized maximum-likelihood estimator which is obtained from concentration inequalities for Bernoulli signals. Numerical experiments confirm the convergence and the robustness of the proposed algorithm.