Paper: arXiv 2409.02025

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.

Complexity vs Empirical Score

  • Math Complexity: 9.2/10
  • Empirical Rigor: 3.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper is mathematically dense, featuring ergodic stochastic control, Hamilton-Jacobi-Bellman equations, and logarithmic regret analysis in continuous time. However, empirical validation is limited to numerical experiments without specific backtest metrics, code, or real-market datasets.

Research Flowchart

  flowchart TD
  A["Research Goal:<br>Regret analysis for learning κ<br>in ergodic Avellaneda-Stoikov model"] --> B["Methodology"]
  
  B --> C["Key Ingredients & Analysis"]
  C --> D["1. HJB Differentiability:<br>Twice differentiable ergodic constant<br>wrt κ (misspecified parameter)"]
  C --> E["2. Learning Theory:<br>Regularised Maximum-Likelihood Estimator<br>for Bernoulli signals"]
  
  B --> F["Computational Process:<br>Simulation of ergodic market<br>with adaptive spreads based on learned κ"]
  
  D & E & F --> G["Main Theoretical Result:<br>Regret upper bound O(ln²T) in expectation"]
  
  H["Numerical Experiments"] --> I["Key Findings:<br>- Convergence of regret<br>- Robustness of algorithm<br>- Validation of theory"]
  
  G --> I