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Mean-Field Games with common Poissonian noise: a Maximum Principle approach

Daniel Hernández-Hernández, Joshué Helí Ricalde-Guerrero

Abstract

The theory of Mean-Field Games is interested in the behaviour of interacting particle systems in which the individual interaction between particles (players) decreases as the size of the population increases. In recent years, it was introduced an interesting structure for this type of games, assuming a correlated continuous source of randomness, which are called Mean-Field Games with Common Noise. In this paper, we extend this concept and provide a precise definition of a Mean-Field Game with Common Poissoninan Noise and its equilibrium. That is, a common self-exciting Poissonian structure is considered within the dynamics of the population. Then, we address the problem of optimization for jump-diffusions with random environments that lies within the definition of the MFG, and develop a stochastic version of the Pontryagin's Maximum Principle to obtain a set of necessary conditions that an optimal control must satisfy. Under additional convexity assumptions it is also shown that these conditions are also sufficient.

Mean-Field Games with common Poissonian noise: a Maximum Principle approach

Abstract

The theory of Mean-Field Games is interested in the behaviour of interacting particle systems in which the individual interaction between particles (players) decreases as the size of the population increases. In recent years, it was introduced an interesting structure for this type of games, assuming a correlated continuous source of randomness, which are called Mean-Field Games with Common Noise. In this paper, we extend this concept and provide a precise definition of a Mean-Field Game with Common Poissoninan Noise and its equilibrium. That is, a common self-exciting Poissonian structure is considered within the dynamics of the population. Then, we address the problem of optimization for jump-diffusions with random environments that lies within the definition of the MFG, and develop a stochastic version of the Pontryagin's Maximum Principle to obtain a set of necessary conditions that an optimal control must satisfy. Under additional convexity assumptions it is also shown that these conditions are also sufficient.
Paper Structure (13 sections, 5 theorems, 143 equations)

This paper contains 13 sections, 5 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Basic notation and preliminaries
  3. Heuristic derivation of the Mean-Field Game
  4. Pontryagin's maximum principle
  5. Hamiltonian system: Definitions and Assumptions
  6. Necessary conditions for optimality
  7. A priori estimates
  8. Necessary conditions through the Hamiltonian system
  9. Sufficient conditions for optimality
  10. Solution to the Mean-Field Game
  11. Examples and applications
  12. Linear-quadratic mean-field game
  13. Portfolio selection with regime switching

Key Result

Lemma 4.6

Let $(\widehat{X},\widehat{\alpha})$ be an optimal pair, and for $\epsilon > 0$ let $(X^\epsilon,\alpha^\epsilon)$ be the corresponding spike variation with respect to $\alpha \in \mathcal{A}$. Then, for each $k=1,\ldots,4$,

Theorems & Definitions (26)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4: hernandez-hernandez_coupled_2023
  • Remark 2.5
  • Remark 3.1
  • Definition 3.2
  • Remark 3.3
  • Remark 4.2
  • Remark 4.3
  • ...and 16 more