Residual U-net with Self-Attention to Solve Multi-Agent Time-Consistent Optimal Trade Execution

ArXiv ID: 2312.09353 “View on arXiv”

Authors: Unknown

Abstract

In this paper, we explore the use of a deep residual U-net with self-attention to solve the the continuous time time-consistent mean variance optimal trade execution problem for multiple agents and assets. Given a finite horizon we formulate the time-consistent mean-variance optimal trade execution problem following the Almgren-Chriss model as a Hamilton-Jacobi-Bellman (HJB) equation. The HJB formulation is known to have a viscosity solution to the unknown value function. We reformulate the HJB to a backward stochastic differential equation (BSDE) to extend the problem to multiple agents and assets. We utilize a residual U-net with self-attention to numerically approximate the value function for multiple agents and assets which can be used to determine the time-consistent optimal control. In this paper, we show that the proposed neural network approach overcomes the limitations of finite difference methods. We validate our results and study parameter sensitivity. With our framework we study how an agent with significant price impact interacts with an agent without any price impact and the optimal strategies used by both types of agents. We also study the performance of multiple sellers and buyers and how they compare to a holding strategy under different economic conditions.

Keywords: Hamilton-Jacobi-Bellman (HJB), backward stochastic differential equation (BSDE), deep residual U-net, optimal trade execution, Almgren-Chriss model, Equities (Trade Execution)

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper employs advanced mathematics including Hamilton-Jacobi-Bellman equations, backward stochastic differential equations, and high-dimensional viscosity solutions, but the summary focuses on theoretical reformulation and neural network architecture without mentioning backtesting, real data, or implementation metrics.

  flowchart TD
    A["Research Goal:<br>Solve Multi-Agent Time-Consistent<br>Optimal Trade Execution"] --> B["Formulate HJB Equation<br>(Almgren-Chriss Model)"]
    B --> C["Reformulate HJB to BSDE<br>for Multiple Agents/Assets"]
    C --> D["Compute Solution via<br>Residual U-Net with Self-Attention"]
    D --> E["Outcomes:"] --> F["Validated Time-Consistent<br>Optimal Control Strategy"]
    D --> E --> G["Interaction Analysis:<br>High vs. Low Price Impact Agents"]
    D --> E --> H["Performance Study:<br>Multiple Sellers/Buyers vs. Holding"]