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

TL;DR

This paper introduces a new deep learning-based numerical method for solving complex McKean-Vlasov FBSDEs with common noise, leveraging elicitability to efficiently approximate solutions without nested simulations.

Contribution

The paper develops a novel approach combining elicitability and deep learning to solve MV-FBSDEs with common noise, avoiding nested Monte Carlo and enabling flexible modeling.

Findings

Accurately solves systemic risk model with analytical solution

Extends to quantile-mediated interactions beyond mean-based measures

Applies to complex economic growth models without closed-form solutions

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.