Stochastic Gradient Descent in the Optimal Control of Execution Costs

ArXiv ID: 2412.12199 “View on arXiv”

Authors: Unknown

Abstract

Bertsimas and Lo’s seminal work laid the groundwork for addressing the implementation shortfall dilemma in institutional investing, emphasizing the significance of market microstructure and price dynamics in minimizing execution costs. However, the ability to derive a theoretical Optimum market order policy is an unrealistic assumption for many investors. This study aims to bridge this gap by proposing an approach that leverages stochastic gradient descent (SGD) to derive alternative solutions for optimizing execution cost policies in dynamic markets where explicit mathematical solutions may not yet exist. The proposed methodology assumes the existence of a mathematically derived optimal solution that is a function of the underlying market dynamics. By iteratively refining strategies using SGD, economists can adapt their approaches over time based on evolving execution strategies. While these SGD-based solutions may not achieve optimality, they offer valuable insights into optimizing policies under complex market frameworks. These results serve as a bridge for economists and mathematicians, facilitating the study of the Optimum policy volatile markets while offering SGD driven implementable policies that closely approximate optimal outcomes within shorter time frames.

Keywords: execution shortfall, stochastic gradient descent, market microstructure, optimization, implementation, Equities

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail
• Why: The paper heavily employs advanced mathematics, including stochastic dynamic programming, Bellman equations, and detailed SGD variants with complex update rules, indicating high mathematical density. While it lacks reported backtest results or real-world data, the methodology is inherently data-driven, requiring iterative refinement of execution strategies based on simulated or historical price dynamics, suggesting strong empirical implementation readiness.

  flowchart TD
    A["Research Goal:<br>Optimize execution costs<br>in dynamic markets"] --> B["Methodology:<br>Stochastic Gradient Descent<br>SGD"]
    B --> C["Data/Inputs:<br>Market Microstructure &<br>Price Dynamics"]
    C --> D["Computational Process:<br>Iterative Policy Refinement<br>via SGD"]
    D --> E["Outcome:<br>Implementable Approximate<br>Optimal Policies"]