Stochastic Delay Differential Games: Financial Modeling and Machine Learning Algorithms

TL;DR

This paper introduces a deep learning-based numerical method to compute Nash equilibria in stochastic delay differential games, addressing high dimensionality in financial and economic models with delays.

Contribution

It develops a novel approach using recurrent neural networks and modified fictitious play to solve complex stochastic delay differential games.

Findings

Successfully computes Nash equilibria in finance-related problems.

Validates method against known solutions and new benchmark problems.

Demonstrates effectiveness in high-dimensional, delayed multi-agent systems.

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.