Stochastic Delay Differential Games: Financial Modeling and Machine Learning Algorithms

ArXiv ID: 2307.06450 “View on arXiv”

Authors: Unknown

Abstract

In this paper, we propose a numerical methodology for finding the closed-loop Nash equilibrium of stochastic delay differential games through deep learning. These games are prevalent in finance and economics where multi-agent interaction and delayed effects are often desired features in a model, but are introduced at the expense of increased dimensionality of the problem. This increased dimensionality is especially significant as that arising from the number of players is coupled with the potential infinite dimensionality caused by the delay. Our approach involves parameterizing the controls of each player using distinct recurrent neural networks. These recurrent neural network-based controls are then trained using a modified version of Brown’s fictitious play, incorporating deep learning techniques. To evaluate the effectiveness of our methodology, we test it on finance-related problems with known solutions. Furthermore, we also develop new problems and derive their analytical Nash equilibrium solutions, which serve as additional benchmarks for assessing the performance of our proposed deep learning approach.

Keywords: Stochastic delay differential games, Recurrent neural networks, Deep learning, Nash equilibrium, Fictitious play, General financial instruments (multi-agent finance)

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 6.0/10
• Quadrant: Lab Rats
• Why: The paper involves dense stochastic calculus, infinite-dimensional HJB equations, and advanced deep learning theory (RNNs, fictitious play), placing it in the high math quadrant. While it validates the method against known solutions and new analytical benchmarks, the lack of real financial data backtesting or specific performance metrics (e.g., Sharpe ratios, transaction costs) limits its empirical rigor compared to purely implementation-focused papers.

  flowchart TD
    A["Research Goal<br>Find NE in Stochastic<br>Delay Differential Games"] --> B["Methodology<br>Deep RL via Recurrent Neural Networks"]
    B --> C["Inputs<br>Finance Problems<br>Known & Analytical NE"]
    C --> D["Computation<br>Modified Fictitious Play<br>with Deep Learning"]
    D --> E{"Evaluation"}
    E --> F["Key Finding 1<br>Effective for High-Dim<br>Finance Problems"]
    E --> G["Key Finding 2<br>Validated on<br>Analytical Benchmarks"]