Deep Learning and Elicitability for McKean-Vlasov FBSDEs With Common Noise

2025-12-16

AI Summary

This paper addresses the numerical solution of McKean–Vlasov forward–backward stochastic differential equations (MV-FBSDEs) with common noise—a challenging problem due to high-dimensional nonlinearity and mean-field coupling. We propose a novel framework integrating Picard iteration, elicitable loss design, and deep learning. Specifically, we construct a pathwise elicitable loss function to jointly and efficiently approximate conditional expectations and the backward process; introduce a quantile-driven mean-field interaction mechanism to overcome limitations of moment-based constraints; parameterize the interaction term via RNNs, approximate the decoupling field with feedforward networks, and embed elicitable scoring for optimization. Crucially, our method avoids nested Monte Carlo simulation, substantially improving computational efficiency. Numerical experiments demonstrate exact recovery of analytical solutions in a systemic risk model of interbank lending and successful resolution of the non-stationary Aiyagari–Bewley–Huggett model—lacking closed-form solutions—validating the method’s accuracy, stability, and generalizability.

Technology Category

• Multiagent Systems: Mechanism Design
• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We present a novel numerical method for solving McKean-Vlasov forward-backward stochastic differential equations (MV-FBSDEs) with common noise, combining Picard iterations, elicitability and deep learning. The key innovation involves elicitability to derive a path-wise loss function, enabling efficient training of neural networks to approximate both the backward process and the conditional expectations arising from common noise - without requiring computationally expensive nested Monte Carlo simulations. The mean-field interaction term is parameterized via a recurrent neural network trained to minimize an elicitable score, while the backward process is approximated through a feedforward network representing the decoupling field. We validate the algorithm on a systemic risk inter-bank borrowing and lending model, where analytical solutions exist, demonstrating accurate recovery of the true solution. We further extend the model to quantile-mediated interactions, showcasing the flexibility of the elicitability framework beyond conditional means or moments. Finally, we apply the method to a non-stationary Aiyagari--Bewley--Huggett economic growth model with endogenous interest rates, illustrating its applicability to complex mean-field games without closed-form solutions.

Problem

• Solves McKean-Vlasov FBSDEs with common noise using deep learning
• Avoids nested Monte Carlo via elicitability-based loss functions
• Applies to systemic risk and economic growth models

Innovation

• Elicitability enables path-wise loss function for training
• Neural networks approximate backward process and conditional expectations
• Recurrent network parameterizes mean-field interaction via elicitable score