阿韦拉内达-斯托伊科夫做市商遍历模型中的对数遗憾

Jialun Cao, David Šiška, Lukasz Szpruch, Tanut Treetanthiploet
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引用次数: 0

摘要

我们分析了在 Avellaneda-Stoikov 市场制造模型的遍历版本中学习流动性接受者的价格敏感性参数$\kappa$所产生的遗憾。我们证明,基于参数的正则化最大似然估计的学习算法在期望值上达到了 $\ln^2 T$ 的后悔值上限。要得到这一结果,我们需要两个关键要素。第一个是汉密尔顿-雅各比-贝尔曼(HJB)方程中关于$\kappa$的遍历常数导数的严格上限。第二个是最大似然估计器的学习率,它是从伯努利信号的集中不等式中得到的。数值实验证实了所提算法的收敛性和鲁棒性。
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Logarithmic regret in the ergodic Avellaneda-Stoikov market making model
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 regularised 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 are tight upper bounds on the derivative of the ergodic constant in the Hamilton-Jacobi-Bellman (HJB) equation with respect to $\kappa$. 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.
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