Paper: arXiv 2601.13493

Authors: Hanchao Liu, Dena Firoozi

Abstract

We extend the results of (Liu and Firoozi, 2025), which develops the theory of linear-quadratic (LQ) mean field games (MFGs) in Hilbert spaces, by incorporating a common noise. This common noise is modeled as an infinite-dimensional Wiener process affecting the dynamics of all agents. In the presence of common noise, the mean-field consistency condition is characterized by a system of coupled forward-backward stochastic evolution equations (FBSEEs) in Hilbert spaces, whereas, in its absence it is represented by coupled forward-backward deterministic evolution equations. We establish the existence and uniqueness of solutions to the coupled linear FBSEEs associated with the LQ MFG framework for small time horizons and prove the $ε$-Nash property of the resulting equilibrium strategy. Furthermore, we establish the well-posedness of these coupled linear FBSEEs for arbitrary finite time horizons. Beyond the specific context of MFGs, our analysis also yields a broader contribution by providing, to the best of our knowledge, the first well-posedness result for a class of infinite-dimensional linear FBSEEs, for which only mild solutions exist, over arbitrary finite time horizons.

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 1.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This is a purely theoretical math paper developing the mathematical foundation of Linear-Quadratic Mean Field Games in infinite-dimensional Hilbert spaces with common noise. It features extreme mathematical complexity (forward-backward stochastic evolution equations, C0-semigroups, Hilbert-Schmidt operators) but lacks any empirical backtesting, financial market data, or implementation code.

Research Flowchart

  flowchart TD
    A[Research Goal: Solve Infinite-Dimensional LQ MFGs with Common Noise] --> B[Inputs: Hilbert spaces, Common noise Wiener process, LQ cost functionals]
    B --> C[Derive Mean-Field Consistency Condition as Coupled Linear FBSEEs]
    C --> D[Prove Well-Posedness of Infinite-Dimensional FBSEEs]
    D -->|Fixed-point argument| E[Small Time Horizons]
    D -->|Monotonicity method| F[Arbitrary Finite Time Horizons]
    E & F --> G[Outcomes: First well-posedness result for class of infinite-D FBSEEs & Proof of ε-Nash equilibrium]

Cited References (150)

  1. Linear Quadratic Nash Systems and Master Equations in Hilbert Spaces (2025) — link
  2. Linear-quadratic optimal control for non-exchangeable mean-field SDEs and applications to systemic risk (2025) — link
  3. Optimal control of heterogeneous mean-field stochastic differential equations with common noise and applications to financial models (2025) — link
  4. Stochastic Optimal Control of Interacting Particle Systems in Hilbert Spaces and Applications (2025) — link
  5. Linear-quadratic stochastic nonzero-sum differential games between graphon teams (2025) — link
  6. Linear Forward-Backward Stochastic Differential Equations (1999)
  7. LQG mean field games with a major agent: Nash certainty equivalence versus probabilistic approach (2022)
  8. Théorie des jeux à champs moyen et applications (2007)
and 142 more
  1. The Master equation in mean field theory (2015)
  2. Simultaneously solving infinitely many LQ mean field games in Hilbert spaces: the power of neural operators — → 2510.20017
  3. A Mean Field Game Approach to Equilibrium Pricing with Market Clearing Condition (2022)
  4. Systemic Risk in Interbanking Networks (2015)
  5. Portfolio liquidation games with self-exciting order flow (2022)
  6. A Mean Field Game of Optimal Portfolio Liquidation (2021)
  7. A mean-field game approach to price formation in electricity markets (2021)
  8. Calculus of variations in Hilbert spaces (2012)
  9. On the minimizers of calculus of variations problems in Hilbert spaces (2015)
  10. A mean field game approach for a class of linear quadratic discrete choice problems with congestion avoidance (2024) — link
  11. Mean field models to regulate carbon emissions in electricity production (2022)
  12. On the backward stochastic R iccati equation in infinite dimensions (2005)
  13. Stochastic equations in infinite dimensions (2014)
  14. Stochastic differential equations in infinite dimensions: with applications to stochastic partial differential equations (2010)
  15. Stochastic optimal control in infinite dimension (2017)
  16. Direct solution of a R iccati equation arising in a stochastic control problem with control and observation on the boundary (1986)
  17. Semilinear evolution equations and their applications (2018)
  18. Analysis in Banach spaces (2016)
  19. Robust Stochastic Games and Systemic Risk (2017)
  20. Robust Mean Field Linear-Quadratic-Gaussian Games with Unknown $L^2$ -Disturbance (2017)
  21. Linear-Quadratic Mean Field Stackelberg Games with State and Control Delays (2017)
  22. Mean Field Stackelberg Games: Aggregation of Delayed Instructions (2015)
  23. Linear quadratic mean field Stackelberg differential games (2018)
  24. $\epsilon$- N ash Equilibria for Partially Observed LQG Mean Field Games with Major Player (2017)
  25. Large Deviations for a Mean Field Model of Systemic Risk (2013)
  26. Mean field game of controls and an application to trade crowding (2018)
  27. The Execution Problem in Finance with Major and Minor Traders: A Mean Field Game Formulation (2017)
  28. Mean‐field games with differing beliefs for algorithmic trading (2020)
  29. A mean-field game approach to equilibrium pricing in solar renewable energy certificate markets (2022)
  30. A mean field game approach to equilibrium pricing with market clearing condition (2022)
  31. Mean field games (2007)
  32. Mean field games (2021)
  33. Probabilistic theory of mean field games with applications I-II (2018)
  34. Mean field games and mean field type control theory (2013)
  35. The master equation and the convergence problem in mean field games:( AMS -201) (2019)
  36. The nucleolus in games with major and minor players (1974)
  37. Large-population LQG games involving a major player: the Nash certainty equivalence principle (2010)
  38. $\epsilon$- Nash mean field game theory for nonlinear stochastic dynamical systems with major and minor agents (2013)
  39. Convex analysis for LQG systems with applications to major–minor LQG mean–field game systems (2020)
  40. A probabilistic approach to mean field games with major and minor players (2016)
  41. An alternative approach to mean field game with major and minor players, and applications to herders impacts (2017)
  42. Large population stochastic dynamic games: C losed-loop McKean-Vlasov systems and the Nash certainty equivalence principle (2006)
  43. Mean field game theory with a partially observed major agent (2016)
  44. $\epsilon$- Nash Equilibria for Major–Minor LQG Mean Field Games With Partial Observations of All Agents (2020)
  45. Economic properties of the risk sensitive criterion for portfolio management (2003)
  46. Risk sensitive asset allocation (2000)
  47. Risk-sensitive control and an optimal investment model (2000)
  48. Optimal stochastic linear systems with exponential performance criteria and their relation to deterministic differential games (1973)
  49. Linear-exponential-quadratic Gaussian control (2013)
  50. Linear exponential quadratic stochastic differential games (2015)
  51. On the optimal control of stochastic systems with an exponential-of-integral performance index (1981)
  52. Risk-sensitive linear/quadratic/Gaussian control (1981)
  53. Bellman equations of risk-sensitive control (1996)
  54. The equivalence between infinite-horizon optimal control of stochastic systems with exponential-of-integral performance index and stochastic differential games (1994)
  55. Risk-sensitive control on an infinite time horizon (1995)
  56. Risk-sensitive mean-field games (2013)
  57. Partially Observed Discrete-Time Risk-Sensitive Mean Field Games (2022)
  58. Risk-sensitive mean-field type control under partial observation (2016)
  59. Brownian motion and stochastic calculus (1991)
  60. Linear and nonlinear functional analysis with applications (2013)
  61. Stochastic controls: Hamiltonian systems and HJB equations (1999)
  62. Optimal control theory: an introduction (2004)
  63. Mean Field Games and systemic risk (2015)
  64. Another proof that convex functions are locally Lipschitz (1974)
  65. A probabilistic weak formulation of mean field games and applications (2015)
  66. Mean-field game strategies for optimal execution (2019)
  67. A mean field games model for cryptocurrency mining (2023)
  68. A mean field game of portfolio trading and its consequences on perceived correlations (2019) — link
  69. Principal agent mean field games in REC markets (2021) — link
  70. Linear quadratic risk-sensitive and robust mean field games (2016)
  71. Risk-sensitive mean field games via the stochastic maximum principle (2019)
  72. Discrete-time risk-sensitive mean-field games (2018) — link
  73. Risk-sensitive zero-sum differential games (2018)
  74. Dynamic noncooperative game theory (1998)
  75. LQG mean field games with a major agent: Nash certainty equivalence versus probabilistic approach (2022)
  76. Linear quadratic mean field social optimization: asymptotic solvability (2019)
  77. Mean-field games with a major player (2018)
  78. Linear-quadratic mean field games with a major player: Nash certainty equivalence versus master equations (2021)
  79. Large Banks and Systemic Risk: Insights from a Mean-Field Game Model — → 2305.17830
  80. A quantilized mean field game approach to energy pricing with application to fleets of plug-in electric vehicles (2019)
  81. Solution formulas for differential Sylvester and Lyapunov equations (2019)
  82. Model simplification and optimal control of stochastic singularly perturbed systems under exponentiated quadratic cost (1996)
  83. Robust designs through risk sensitivity: An overview (2021)
  84. A new risk-sensitive maximum principle (2005)
  85. Introduction to stochastic integration (1990)
  86. Dynamic programming approach to stochastic evolution equations (1979)
  87. Stability of semilinear stochastic evolution equations (1982)
  88. The stochastic linear quadratic optimal control problem on Hilbert spaces: The case of non-analytic systems (2023)
  89. The stochastic linear quadratic control problem with singular estimates (2017)
  90. Mathematical control theory for stochastic partial differential equations (2021)
  91. Stochastic evolution equations (2008)
  92. Linear-quadratic mean field games (2016)
  93. Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized $\varepsilon$- Nash equilibria (2007)
  94. Social optima in mean field LQG control: centralized and decentralized strategies (2012)
  95. Mean field game with delay: a toy model (2018)
  96. Systemic risk and stochastic games with delay (2018)
  97. Linear operators, part 1: general theory (1988)
  98. Stochastic partial differential equations with L 'e vy noise: An evolution equation approach (2007)
  99. Linear-quadratic mean field Stackelberg games with state and control delays (2017)
  100. Linear–quadratic mean-field game for stochastic delayed systems (2018)
  101. Some remarks on the R iccati equation arising in an optimal control problem with state-and control-dependent noise (1992)
  102. Well-posedness of stochastic R iccati equations and closed-loop solvability for stochastic linear quadratic optimal control problems (2019)
  103. Linear Stochastic Processes on Networks and Low Rank Graph Limits (2023)
  104. Graphon Field Tracking Games with Discrete Time Q -noise (2023)
  105. Linear stochastic graphon systems with Q -space noise (2022)
  106. Itô’s lemma in infinite dimensions (1970)
  107. Measure theory (2013)
  108. Direct solution of a R iccati equation arising in stochastic control theory (1984)
  109. Stochastic LQ Control and Associated R iccati Equation of PDE s Driven by State-and Control-Dependent White Noise (2022)
  110. Introductory functional analysis with applications (1991)
  111. Linear-Quadratic Mean Field Games in Hilbert Spaces (2025)
  112. Explicit solutions of some linear-quadratic mean field games (2012)
  113. LQG Risk-Sensitive Single-Agent and Major-Minor Mean Field Game Systems: A Variational Framework — → 2305.15364
  114. Quadratic Optimal Control of Graphon Q -noise Linear Systems (2024)
  115. Optimal control of path-dependent McKean–Vlasov SDEs in infinite-dimension (2023)
  116. Semigroups of linear operators and applications (2017)
  117. Functional Analysis (1991)
  118. Linear-quadratic mean field games of controls with non-monotone data (2023)
  119. Risk-sensitive mean field games with major and minor players (2023)
  120. Adapted solution of a backward semilinear stochastic evolution equation (1991)
  121. It ^o ’s formula in UMD Banach spaces and regularity of solutions of the Zakai equation (2008)
  122. Mean field games with common noise (2016)
  123. Singular integral operators: a martingale approach (1990)
  124. Spaces of operator-valued functions measurable with respect to the strong operator topology (2010)
  125. Selection of equilibria in a linear quadratic mean-field game (2019)
  126. Linear Quadratic Mean Field Type Control and Mean Field Games with Common Noise, with Application to Production of an Exhaustible Resource (2016)
  127. Wellposedness of Mean Field Games with Common Noise under a Weak Monotonicity Condition (2016)
  128. Closed-loop convergence for mean field games with common noise (2023)
  129. On mean field games with common noise and McKean-Vlasov SPDEs (2019)
  130. Translation invariant mean field games with common noise (2015)
  131. A general characterization of the mean field limit for stochastic differential games (2016)
  132. Uniqueness for linear-quadratic mean field games with common noise (2018)
  133. Linear-quadratic-Gaussian mean-field-game with partial observation and common noise (2021)
  134. On Mean Field Games in Infinite Dimension (2024) — link
  135. Controlled Markov processes and viscosity solutions (2006)
  136. Risk-sensitive mean field games with common noise: a theoretical study with applications to interbank markets — → 2403.03915
  137. A class of hybrid LQG mean field games with state-invariant switching and stopping strategies (2022)
  138. Exploratory LQG mean field games with entropy regularization (2022)
  139. Stochastic Models for Prices Dynamics in Energy and Commodity Markets: An Infinite-Dimensional Perspective (2023)
  140. A stochastic partial differential equation model for limit order book dynamics (2021)
  141. Equilibrium in functional stochastic games with mean-field interaction — → 2306.05433
  142. Hilbert space-valued LQ mean field games: An infinite-dimensional analysis (2025)

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