Deep Signature Approach for McKean-Vlasov FBSDEs in a Random Environment

TL;DR

This work addresses the challenge of solving MV-FBSDEs in random environments with common noise and full distribution dependence by introducing a deep learning fictitious-play framework. The method combines path-signature representations of the conditional law with deep BSDE solvers, learning distribution-dependent coefficients via supervised regression and iteratively updating decoupling fields. Under Lipschitz and growth conditions, the authors prove convergence of the fictitious-play scheme up to a supervised-learning error and demonstrate effectiveness through three numerical experiments, including a nonlinear flocking model and a high-dimensional MV-FBSDE with analytical benchmarks. The approach enables scalable, high-dimensional Mean-Field Game calculations with general distributional interactions and common shocks, offering a practical tool for systems with strong collective effects and shared randomness.

Abstract

Mean-field games with common noise provide a powerful framework for modeling the collective behavior of large populations subject to shared randomness, such as systemic risk in finance or environmental shocks in economics. These problems can be reformulated as McKean-Vlasov forward-backward stochastic differential equations (MV-FBSDEs) in a random environment, where the coefficients depend on the conditional law of the state given the common noise. Existing numerical methods, however, are largely limited to cases where interactions depend only on expectations or low-order moments, and therefore cannot address the general setting of full distributional dependence. In this work, we introduce a deep learning-based algorithm for solving MV-FBSDEs with common noise and general mean-field interactions. Building on fictitious play, our method iteratively solves conditional FBSDEs with fixed distributions, where the conditional law is efficiently represented using signatures, and then updates the distribution through supervised learning. Deep neural networks are employed both to solve the conditional FBSDEs and to approximate the distribution-dependent coefficients, enabling scalability to high-dimensional problems. Under suitable assumptions, we establish convergence in terms of the fictitious play iterations, with error controlled by the supervised learning step. Numerical experiments, including a distribution-dependent mean-field game with common noise, demonstrate the effectiveness of the proposed approach.

Figures (7)

• Figure 1: Mean absolute error (MAE) over time for Archi. 1 (linear functional of the truncated signature) and Archi. 2 (feedforward neural network with truncated signature input), in dimensions 1, 5, and 10. Solid and dashed lines denote the average MAE across 5 independent runs; shaded areas show $\pm$ standard deviation.
• Figure 2: Mean absolute error (MAE) over time for signature truncation orders $M=2,3,4$ in dimensions $d=p = 1, 5, 10$. Solid and dashed lines represent the average MAE across 5 independent runs; shaded areas indicate $\pm$ one standard deviation. Lower truncation orders yield better performance, with $M=2$ achieving the lowest error.
• Figure 3: Mean Euclidean Error (MEE) over time for $d = p = 5$ with varying numbers of fictitious play iterations. Shaded regions represent $\pm 1$ standard deviation over five independent runs.
• Figure 4: Comparison of analytical solutions (dashed line) and numerical solutions (solid lines) for $d = q = 5$ obtained after 30 fictitious play rounds over five independent runs.
• Figure 5: Training loss for Cucker-Smale model with different $\beta$. Left: the loss of supervised learning of $m_4$ in Step 2. Right: the loss of Deep BSDE in Step 3.

Theorems & Definitions (12)

• Remark 2.1
• Remark 2.2: Existence of solutions and relation to MFG
• Definition 2.4: Signature
• Definition 2.5: Log-Signature
• Lemma 4.2
• proof
• Theorem 4.3
• proof
• Lemma 4.4
• proof