Beyond separability: convergence rate of vanishing viscosity approximations to mean field games via FBSDE stability
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916946494095360 |
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| author | Yu, Winston Du, Qiang Tang, Wenpin |
| author_facet | Yu, Winston Du, Qiang Tang, Wenpin |
| contents | This paper studies the vanishing viscosity approximation to mean field games (MFGs) in $\mathbb{R}^d$ with a nonlocal and possibly non-separable Hamiltonian. We prove that the value function converges at a rate of $\mathcal{O}(β)$, where $β^2$ is the diffusivity constant, which matches the classical convergence rate of vanishing viscosity for Hamilton-Jacobi (HJ) equations. The same rate is also obtained for the approximation of the distribution of players as well as for the gradient of the value function. The proof is a combination of probabilistic and analytical arguments by first analyzing the forward-backward stochastic differential equation associated with the MFG, and then applying a general stability result for HJ equations. Applications of our result to $N$-player games, mean field control, and policy iteration for solving MFGs are also presented. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_18529 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Beyond separability: convergence rate of vanishing viscosity approximations to mean field games via FBSDE stability Yu, Winston Du, Qiang Tang, Wenpin Optimization and Control Analysis of PDEs Probability 35Q89, 49L25, 60H10, 91A15, 93E20 This paper studies the vanishing viscosity approximation to mean field games (MFGs) in $\mathbb{R}^d$ with a nonlocal and possibly non-separable Hamiltonian. We prove that the value function converges at a rate of $\mathcal{O}(β)$, where $β^2$ is the diffusivity constant, which matches the classical convergence rate of vanishing viscosity for Hamilton-Jacobi (HJ) equations. The same rate is also obtained for the approximation of the distribution of players as well as for the gradient of the value function. The proof is a combination of probabilistic and analytical arguments by first analyzing the forward-backward stochastic differential equation associated with the MFG, and then applying a general stability result for HJ equations. Applications of our result to $N$-player games, mean field control, and policy iteration for solving MFGs are also presented. |
| title | Beyond separability: convergence rate of vanishing viscosity approximations to mean field games via FBSDE stability |
| topic | Optimization and Control Analysis of PDEs Probability 35Q89, 49L25, 60H10, 91A15, 93E20 |
| url | https://arxiv.org/abs/2505.18529 |