| Abstract: |
| Mean-field forward-backward stochastic differential equations (FBSDEs) serve as a core mathematical framework in mean-field game theory with broad applications in finance and large population systems. By leveraging the connection between FBSDEs and high-dimensional partial differential equations, deep learning algorithms can effectively circumvent the curse of dimensionality. However, existing numerical methods train neural network parameters separately on each time subinterval, causing parameter proliferation as time partitions become finer and leading to unstable computational accuracy.
To address this issue, we note that under Markovian coefficient conditions, there exists a unique function u such that $Y_t = u(t, X_t)$ over the entire time interval, enabling consistent parameter training and significantly improving numerical accuracy. This result also holds when mean-field terms are expressed in distribution form, provided that certain regularity conditions are satisfied. Furthermore, the convergence from theoretical solutions to discretized equations is studied, and the convergence of neural network solutions with respect to the proposed loss function is analyzed. Several numerical examples are provided to validate the effectiveness of the method. |
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