Abstract
Financial markets and more generally macro-economic models involve a large number of individuals interacting through variables such as prices resulting from the aggregate behavior of all the agents. Mean field games have been introduced to study Nash equilibria for such problems in the limit when the number of players is infinite. The theory has been extensively developed in the past decade, using both analytical and probabilistic tools, and a wide range of applications have been discovered, from economics to crowd motion. More recently the interaction with machine learning has attracted a growing interest. This chapter reviews the literature on the interplay between mean field games and deep learning, with a focus on three families of methods. A special emphasis is given to financial applications.
Summary
Three distinct deep learning approaches for MFG/MFC, each with different trade-offs for implementation:
Method 1: Direct SGD for MKV Control (Algorithm 1). Parameterize the control alpha as a neural network alpha_theta(t, x). Simulate N interacting particles with empirical mean-field distribution. Discretize time. Minimize the cost functional J^{N,Delta t}(theta) directly via SGD — no PDE, no FBSDE, no optimality conditions needed. This is the simplest to implement. Applied to a price impact model with transaction costs (Section 2.2): traders with quadratic transaction costs c_alpha/2 * alpha^2 and holding costs c_X/2 * X^2 interact through aggregate trading rate. Parameters: N=2000 particles, N_T=50 time steps, sigma=0.5, T=1.
Method 2: Deep BSDE Shooting for MKV FBSDEs (Algorithm 2). For MFG equilibria characterized by coupled FBSDEs, parameterize the initial value Y_0 = y_{0,theta} and the volatility of Z = z_omega as neural networks. Simulate forward, compute terminal mismatch loss J_{FBSDE} = E|Y_T - G(X_T, L(X_T))|^2. SGD on (theta, omega). Applied to systemic risk MFG (Section 3.2) — the same inter-bank model used in the DFP papers. Handles common noise by conditioning on the filtration of W^0.
Method 3: Deep Galerkin Method for Mean Field PDEs (Algorithm 3). For MFG PDE systems (coupled KFP + HJB), parameterize density m = m_{theta_1} and value u = u_{theta_2} as two neural networks. Loss = L^{KFP}(m,u) + L^{HJB}(m,u) where each term is the L^2 PDE residual plus boundary condition penalties (eqs. 17-18). Collocation points sampled uniformly. Applied to crowded trade model with transaction costs (Section 4.2) where kappa > 0 is the quadratic transaction cost parameter.
Key Contributions
- Unified survey of three complementary deep learning approaches for MFG/MFC
- All three illustrated on financial applications with transaction costs or price impact
- Method 1 requires no knowledge of optimality conditions — just simulate and optimize
- Method 2 extends the deep BSDE method to McKean-Vlasov FBSDEs
- Method 3 (deep Galerkin method) is the most general but hardest to tune
Methodology
The methods target different formulations of the same problem: Method 1 works at the control level (no PDE), Method 2 at the FBSDE level (probabilistic), Method 3 at the PDE level (analytical). For implementation, Method 1 is simplest (just differentiate the cost), Method 2 leverages the BSDE structure for efficiency, Method 3 is most flexible but requires careful tuning of loss weights C^{KFP}, C_0^{KFP}, C^{HJB}, C_T^{HJB}.
Key Findings
- Method 1 (direct SGD) matches semi-explicit ODE solutions for price impact model
- Method 2 (BSDE shooting) approximation is better for X^i than Y^i (errors accumulate in forward Y equation)
- Method 3 (DGM) loss weight selection is critical — poor weights lead to trivial solutions
- Common noise makes all methods harder; Method 2 handles it by conditioning
- Transaction cost examples appear in both Methods 1 and 3
Important References
- Probabilistic Theory of Mean Field Games with Applications I — Carmona, Delarue (2018), theoretical foundation
- Mean Field Games and Systemic Risk — Carmona, Fouque, Sun (2015), the systemic risk model used as benchmark
- Convergence Analysis of Machine Learning Algorithms for MFG and MFC — Carmona, Lauriere (2019/2021), convergence proofs