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Mean Field Game of Mutual Holding with common noise

Leila Bassou, Mao Fabrice Djete, Nizar Touzi

Abstract

We consider the mean field game of cross--holding introduced in \citeauthor*{DjeteTouzi} \cite{DjeteTouzi} in the context where the equity value dynamics are affected by a common noise. In contrast with \cite{DjeteTouzi}, the problem exhibits the standard paradigm of mean--variance trade off. Our crucial observation is to search for equilibrium solutions of our mean field game among those models which satisfy an appropriate notion of no--arbitrage. Under this condition, it follows that the representative agent optimization step is reduced to a standard portfolio optimization problem with random endowment.

Mean Field Game of Mutual Holding with common noise

Abstract

We consider the mean field game of cross--holding introduced in \citeauthor*{DjeteTouzi} \cite{DjeteTouzi} in the context where the equity value dynamics are affected by a common noise. In contrast with \cite{DjeteTouzi}, the problem exhibits the standard paradigm of mean--variance trade off. Our crucial observation is to search for equilibrium solutions of our mean field game among those models which satisfy an appropriate notion of no--arbitrage. Under this condition, it follows that the representative agent optimization step is reduced to a standard portfolio optimization problem with random endowment.
Paper Structure (11 sections, 8 theorems, 109 equations)

This paper contains 11 sections, 8 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. The one--period mean field game of cross--holding
  3. No--arbitrage (NA)
  4. The mean--variance criterion
  5. Continuous time MFG of cross--holding
  6. Arbitrage free random environment
  7. The reduced mean field game
  8. Black--Scholes idiosyncratic dynamics examples
  9. The logarithmic utility case
  10. The power utility case
  11. Appendix: proofs in the one--period MFG

Key Result

Lemma 2.3

Under the no--arbitrage condition condition_K_NA, the set $K$ is closed in $\mathbb{L}^1$.

Theorems & Definitions (19)

  • Definition 2.1: One period mean field game of cross--holding
  • Definition 2.2: No--arbitrage (NA)
  • Lemma 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Example 2.6: cross--holding depending on the second argument only
  • Example 2.7: Separable form
  • Definition 3.2
  • Example 3.3
  • Definition 3.4
  • ...and 9 more