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

ArXiv ID: 2512.14967 “View on arXiv”

Authors: Felipe J. P. Antunes, Yuri F. Saporito, Sebastian Jaimungal

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.

Keywords: McKean-Vlasov, Forward-Backward SDE, Deep Learning, Systemic Risk, Mean-Field Games

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail
• Why: The paper employs advanced mathematical tools like McKean-Vlasov FBSDEs, elicitability theory, and pathwise loss functions, indicating high math complexity. It demonstrates empirical rigor with neural network implementations, validation on models with analytical solutions, and application to complex economic models, showing backtest-ready methodology.

  flowchart TD
    A["Research Goal: Develop & Validate Deep Learning Method for MV-FBSDEs with Common Noise"] --> B["Methodology: Picard Iterations + Elicitability + Deep Learning"]
    B --> C{"Data/Input: MV-FBSDE Formulations<br/>Systemic Risk Model / Aiyagari-Bewley-Huggett Model"}
    C --> D["Computational Process: Train NN to approximate Backward Process & Conditional Expectations<br/>Loss: Path-wise Elicitable Score"]
    D --> E{"Outcome: Accurate Solution Recovery<br/>Validated on Analytical Systemic Risk Model"}
    E --> F["Key Finding: Flexible Framework<br/>Extends to Quantile Interactions & Non-Stationary Economic Models"]
    F --> G["Conclusion: Efficient Method for Complex Mean-Field Games"]