Logarithmic regret in the ergodic Avellaneda-Stoikov market making model

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 regularised maximum-likelihood estimator for the parameter achieves the regret upper bound of order ln² T in expectation. To obtain the result we need two key ingredients. The first are tight upper bounds on the derivative of the ergodic constant in the Hamilton-Jacobi-Bellman (HJB) equation with respect to κ. The second is the learning rate of the maximum-likelihood estimator which is obtained from concentration inequalities for Bernoulli signals. Numerical experiment confirms the convergence and the robustness of the proposed algorithm.