Logarithmic regret in the ergodic Avellaneda-Stoikov market making model
Logarithmic regret in the ergodic Avellaneda-Stoikov market making model ArXiv ID: 2409.02025 “View on arXiv” Authors: Unknown Abstract We analyse the regret arising from learning the price sensitivity parameter $κ$ 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 $κ$, 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. ...