Paper: arXiv 2301.05157

Authors: Eyal Neuman, Yufei Zhang

Abstract

We consider a class of learning problems in which an agent liquidates a risky asset while creating both transient price impact driven by an unknown convolution propagator and linear temporary price impact with an unknown parameter. We characterize the trader’s performance as maximization of a revenue-risk functional, where the trader also exploits available information on a price predicting signal. We present a trading algorithm that alternates between exploration and exploitation phases and achieves sublinear regrets with high probability. For the exploration phase we propose a novel approach for non-parametric estimation of the price impact kernel by observing only the visible price process and derive sharp bounds on the convergence rate, which are characterised by the singularity of the propagator. These kernel estimation methods extend existing methods from the area of Tikhonov regularisation for inverse problems and are of independent interest. The bound on the regret in the exploitation phase is obtained by deriving stability results for the optimizer and value function of the associated class of infinite-dimensional stochastic control problems. As a complementary result we propose a regression-based algorithm to estimate the conditional expectation of non-Markovian signals and derive its convergence rate.

Complexity vs Empirical Score

  • Math Complexity: 9.2/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents extremely dense and advanced mathematical machinery, combining non-parametric Tikhonov regularization, infinite-dimensional stochastic control, and sublinear regret analysis for optimal execution with price impact. However, it is primarily a theoretical contribution with little to no empirical backtesting or real-world data validation. Novelty in combining propagator estimation with reinforcement-learning-style regret bounds is very high.

Research Flowchart

  flowchart TD
    A[Research Goal: Sublinear Regret in Optimal Liquidation with Unknown Impact] --> B[Inputs: Visible Price Process & Non-Markovian Signals]
    
    B --> C[Exploration: Non-Parametric Kernel Estimation via Tikhonov Regularization]
    B --> D[Exploitation: Infinite-Dimensional Stochastic Control Optimization]
    
    C --> E[Regression-Based Signal Conditional Expectation Estimation]
    
    E --> F[Combined Trading Algorithm: Alternating Exploration & Exploitation]
    
    F --> G[Outcomes: Sharp Convergence Rates & High-Probability Sublinear Regret]