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

Felipe J. P. Antunes; Yuri F. Saporito; Sebastian Jaimungal

cs.LGDec 16, 2025

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

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

Organizations: School of Applied Mathematics, Getulio Vargas Foundation, Brazil. · Oxford-Man Institute for Quantitative Finance, University of Oxford

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 hybrid feedforward and recurrent 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.

Figures & tables

X^{n+1}_{t}=X_{0}+\int_{0}^{t}\mu(u,X^{n}_{u},Y_{u},Z_{u},Z^{0}_{u},S_{u})\,\,{{\rm d}u}+\int_{0}^{t}\sigma(u,X^{n}_{u})\,{{\rm d}W}_{u}+\int_{0}^{t}\sigma_{0}(u,X^{n}_{u})\,{{\rm d}W}^{0}_{u};

Algorithm 2 Picard iteration of forward SDE with fixed

(Y,Z,Z^{0},m)

\hskip-28.45274pt\mathcal{L}(\theta)=\frac{1}{IP}\sum_{{t\in\mathcal{T}}}\sum_{i=1}^{I}p_{t}\left(U_{\theta}(t,X_{t}^{(m_{i})},(W_{s}^{0(m_{i})})_{u\leq t,u\in\mathcal{T}})-G(X_{T}^{(m_{i})},S_{T}^{(m_{i})})-\sum_{u\in{\mathcal{T}},u\geq t}f_{u}^{(m_{i})}\,\,\Delta t\right)^{2};

Algorithm 4 Estimating

Y

through elicitability with fixed

(X,Z,Z^{0},m)

\mathcal{L}(\theta)=\frac{1}{IN}\sum_{{t\in\mathcal{T}}\setminus\{T\}}\sum_{i=1}^{I}\left(V_{\theta}(t,X_{t}^{(m_{i})},(W_{s}^{0(m_{i})})_{s\leq t,s\in\mathcal{T}})-\left(\frac{\Delta Y^{(m_{i})}_{t}}{\Delta t}+f_{t}^{(m_{i})}\right)\,\Delta W_{t}^{(m_{i})}\right)^{2};

Algorithm 5 Estimating

Z

and

Z^{0}

through elicitability with fixed

(X,Y,m)

Panel of Figure 1 — Figure 1. Architecture of the hybrid approximator.

$d$	$K$	$X$		$Y$		$Z$		$Z^{0}$		$S$
$d=1$	$k=5$	$3.54$ (	1.99 )	$8.22$ (	5.97 )	$16.80$ (	11.2 )	$19.70$ (	3.54 )	$3.82$ (	2.03 )
	$k=50$	$0.54$ (	0.46 )	$1.08$ (	1.00 )	$2.08$ (	2.18 )	$3.84$ (	1.84 )	$0.45$ (	0.56 )
	$k=100$	$0.59$ (	0.44 )	$1.53$ (	0.94 )	$1.71$ (	1.30 )	$1.76$ (	1.11 )	$0.36$ (	0.45 )
	$k=250$	$0.41$ (	0.34 )	$0.81$ (	0.81 )	$1.59$ (	1.13 )	$0.60$ (	0.60 )	$0.47$ (	0.44 )
$d=5$	$k=5$	$11.10$ (	6.94 )	$31.50$ (	17.2 )	$45.20$ (	25.0 )	$45.40$ (	7.45 )	$12.60$ (	8.18 )
	$k=50$	$2.29$ (	1.35 )	$4.95$ (	2.92 )	$3.67$ (	3.09 )	$5.17$ (	2.21 )	$3.01$ (	2.08 )

Table 1. MEE results (

\times 10^{-2}

) for different values of

K

and dimensions. In parentheses, the standard deviation of the Euclidean norm of errors by path and time.

Figure 2. Comparison between analytic solution and approximated solution on two sample paths for the systemic risk with common noise experiment. Solid lines are the numerical solutions, and dotted lines represent the analytical solution. Each color pair (orange and blue, green and red) represents a particular path of each variable of the system.

Figure 3. The same sample path at iterations 1,2,4,8,16 and 32. As iteration increases, the color becomes darker.

Figure 4. Modified systemic risk model where interaction is mediated through the

60\%

quantile. Respective paths (subjected to the same Brownian motion) of the original, mean interaction systemic risk model are shown in lighter shades. Note that the

X_{t}

variable is greater in the quantile interaction model.

Panel of Figure 7 — Figure 7. Marginal propensity to consume as a function of $K_{t}$ and $r_{t}$ , for time $t=0.5$ .

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A Posteriori Error Analysis for Decoupled Neural Approximations of Fully Coupled FBSDEs with Control Mismatch

Xichuan Zhang

Intelligent Game and Decision Lab, Beijing 100091, China

Stochastic Differential Equations Neural PDE Solvers

Open paper AskPDF arXiv

This paper develops an a posteriori error analysis framework for decoupled neural approximations of fully coupled forward--backward stochastic differential equations (FBSDEs). It provides an a posteriori error-analysis for the idealized discrete adapted trajectory. The main feature of the proposed formulation is the use of an auxiliary control process in the forward coefficients, which may differ from the backward component approximated by the neural network. This decoupling is useful in practical deep learning implementations, but it creates a control mismatch that must be included in the error analysis. We first establish a continuous-time stability estimate for fully coupled FBSDEs under perturbations of the drift, diffusion, generator, terminal condition, and auxiliary control input. We then transfer this estimate to the discrete-time setting and derive computable a posteriori error bounds depending only on the terminal defect, the pathwise residual, and the control mismatch. When the auxiliary control is identified with the backward approximation, the mismatch term vanishes and the bound reduces to the standard two-term form. Numerical experiments on a linear--quadratic FBSDE with an explicit reference solution and a multidimensional Burgers-type FBSDE without a reference solution illustrate the diagnostic role of the proposed indicators and the contribution of the mismatch penalty to the consistency and reproducibility of the numerical approximations.

Abstract

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