Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Switching to a Green and sustainable finance setting: a mean field game approach

Anna Aksamit, Kaustav Das, Ivan Guo, Kihun Nam, Zhou Zhou

TL;DR

This work tackles a Green and sustainable finance problem modeled as a mean field game with a major regulator and a continuum of minor firms. By relaxing the dependence of the Hamiltonian minimisers, the authors extend the stochastic maximum principle to allow richer regulator–firm interactions, enabling explicit Nash equilibria in ESG-inspired examples. They develop a MmMFG framework, prove necessary and, under convexity, sufficient SMP conditions, and address finite‑player approximations via $\varepsilon_N$‑Nash equilibria; they further enlarge the state space to accommodate joint distributions, linking to extended mean field game methods. The applications demonstrate tractable, explicit equilibria in simple cases and illustrate the framework’s potential for tackling more complex, policy-relevant interactions in large populations, with numerical methods proposed for future work.

Abstract

We consider a continuum of carbon-emitting firms who seek to maximise their stock price, and a regulator (e.g., Government) who wishes for the economy to flourish, whilst simultaneously punishing firms who behave non-green. Interpreting the regulator as a major player and the firms as the minor players, we model this setting through a mean field game with major and minor players. We extend the stochastic maximum principle derived by Carmona & Zhu [A probabilistic approach to mean field games with major and minor players. Annals of Applied Probability, 2016, 94, 745--788] by relaxing the assumptions on the forms of the minimisers for the Hamiltonians, allowing them to depend on more arguments. This allows the major and representative minor player to interact in a more natural fashion, thereby permitting us to consider more realistic models for our green and sustainable finance problem. Through our stochastic maximum principle, we derive explicit Nash equilibria for a number of examples.

Switching to a Green and sustainable finance setting: a mean field game approach

TL;DR

This work tackles a Green and sustainable finance problem modeled as a mean field game with a major regulator and a continuum of minor firms. By relaxing the dependence of the Hamiltonian minimisers, the authors extend the stochastic maximum principle to allow richer regulator–firm interactions, enabling explicit Nash equilibria in ESG-inspired examples. They develop a MmMFG framework, prove necessary and, under convexity, sufficient SMP conditions, and address finite‑player approximations via ‑Nash equilibria; they further enlarge the state space to accommodate joint distributions, linking to extended mean field game methods. The applications demonstrate tractable, explicit equilibria in simple cases and illustrate the framework’s potential for tackling more complex, policy-relevant interactions in large populations, with numerical methods proposed for future work.

Abstract

We consider a continuum of carbon-emitting firms who seek to maximise their stock price, and a regulator (e.g., Government) who wishes for the economy to flourish, whilst simultaneously punishing firms who behave non-green. Interpreting the regulator as a major player and the firms as the minor players, we model this setting through a mean field game with major and minor players. We extend the stochastic maximum principle derived by Carmona & Zhu [A probabilistic approach to mean field games with major and minor players. Annals of Applied Probability, 2016, 94, 745--788] by relaxing the assumptions on the forms of the minimisers for the Hamiltonians, allowing them to depend on more arguments. This allows the major and representative minor player to interact in a more natural fashion, thereby permitting us to consider more realistic models for our green and sustainable finance problem. Through our stochastic maximum principle, we derive explicit Nash equilibria for a number of examples.

Paper Structure

This paper contains 20 sections, 2 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Prelimininaries
  3. Model framework
  4. Finite player setting
  5. A stochastic maximum principle
  6. A necessary stochastic maximum principle
  7. A sufficient stochastic maximum principle
  8. Approximate Nash equilibria for the finite player game
  9. Enlarging the state space
  10. Applications to Green and sustainable Finance
  11. Example 1
  12. Example 2
  13. Example 3
  14. Conclusion
  15. L differentiation
  16. ...and 5 more sections

Key Result

Theorem 3.1

Enforce Assumptions ass:regularity (ass:regularity) and ass:necessarysmp (ass:necessarysmp). Let StepsMmMFG:Step1MmMFG:Step2 of the MmMFGstrategy be fulfilled, and denote by $(\hat{\alpha}^\mu_0, \hat{\alpha}^\mu)$ the associated Nash equilibrium. Let $(\hat{\underline{X}}, \hat{\underline{P}}, \hat for all $\alpha_0 \in A_0$, $\mathrm{d} t$ a.e. and for all $\alpha \in A$, $\mathrm{d} \mathbb{P}

Theorems & Definitions (14)

  • Remark 2.1: Matrix notation
  • Remark 2.2
  • Definition 2.1: $\varepsilon$-Nash equilibrium
  • Theorem 3.1: Necessary stochastic maximum principle for MmMFG strategy
  • proof
  • Remark 3.1
  • Theorem 3.2: Sufficient stochastic maximum principle for MmMFG strategy
  • proof
  • Remark 3.2
  • Definition A.1: Lifted function
  • ...and 4 more