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<p>In the second part of our work, we aim to establish the global-in-time well-posedness of classical solution of the master equations associated with general mean field games studied in Part I, which is beyond the specific linear-quadratic setting, provided the mean field sensitivity effect is not too large. We characterize the gradient of the value function by the backward process of the forward-backward stochastic differential equations (FBSDEs) introduced in Part I. Then we study the higher regularity of Jacobian flows of the FBSDEs in the state and measure variables so as to establish classical well-posedness of the master equation on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {R}^d\)</EquationSource> </InlineEquation>. As far as we know, it is the first work to investigate the master equations, with general cost functions having quadratic growth and allowing non-convexity in the state variable, under the small mean field effect. Our current approach directly imposes the structural assumptions (most notably, the small mean field sensitivity effect) on the cost functions, which provides the following advantages: (i) the structural conditions imposed in this work are easily verified and less demanding on the assumptions of the cost functions; (ii) we illustrate how the displacement monotonicity should be formulated when the assumptions are imposed on the cost functions instead of the Hamiltonian; and (iii) we provide an accurate lifespan, which may not be that small in many circumstances, for the local-in-time existence when the mean field sensitivity effect is relatively large, the cost functions are not convex in the state variable or we do not have the monotonicity of cost functions.</p>

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A Control Theoretical Approach to Mean Field Games. Part II: Global Well-Posedness of Master Equations

  • 期刊论文

  • 发表日期：

    2026年3月5日

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  • Alain Bensoussan,
  • Ho Man Tai,
  • Tak Kwong Wong,
  • Sheung Chi Phillip Yam

摘要

In the second part of our work, we aim to establish the global-in-time well-posedness of classical solution of the master equations associated with general mean field games studied in Part I, which is beyond the specific linear-quadratic setting, provided the mean field sensitivity effect is not too large. We characterize the gradient of the value function by the backward process of the forward-backward stochastic differential equations (FBSDEs) introduced in Part I. Then we study the higher regularity of Jacobian flows of the FBSDEs in the state and measure variables so as to establish classical well-posedness of the master equation on \(\mathbb {R}^d\) . As far as we know, it is the first work to investigate the master equations, with general cost functions having quadratic growth and allowing non-convexity in the state variable, under the small mean field effect. Our current approach directly imposes the structural assumptions (most notably, the small mean field sensitivity effect) on the cost functions, which provides the following advantages: (i) the structural conditions imposed in this work are easily verified and less demanding on the assumptions of the cost functions; (ii) we illustrate how the displacement monotonicity should be formulated when the assumptions are imposed on the cost functions instead of the Hamiltonian; and (iii) we provide an accurate lifespan, which may not be that small in many circumstances, for the local-in-time existence when the mean field sensitivity effect is relatively large, the cost functions are not convex in the state variable or we do not have the monotonicity of cost functions.