Logarithmic regret in the ergodic Avellaneda-Stoikov market making model

TL;DR

This paper studies the regret in learning the price sensitivity parameter in an ergodic market making model, showing that a maximum-likelihood based algorithm achieves a regret of order ^2, with theoretical and numerical validation.

Contribution

It establishes the ^2 regret bound for a learning algorithm in the ergodic Avellaneda-Stoikov model, combining HJB analysis and concentration inequalities.

Findings

The regret upper bound is of order ^2 in expectation.

The proposed algorithm is robust and converges in numerical experiments.

Key mathematical tools include differentiability of the ergodic constant and concentration inequalities.

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.