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Optimal consumption and investment under relative performance criteria with Epstein-Zin utility

Jodi Dianetti, Frank Riedel, Lorenzo Stanca

Abstract

We consider the strategic interaction of traders in a continuous-time financial market with Epstein-Zin-type recursive intertemporal preferences and performance concerns. We derive explicitly an equilibrium for the finite player and the mean-field version of the game, based on a study of geometric backward stochastic differential equations of Bernoulli type that describe the best replies of traders. Our results show that Epstein-Zin preferences can lead to substantially different equilibrium behavior.

Optimal consumption and investment under relative performance criteria with Epstein-Zin utility

Abstract

We consider the strategic interaction of traders in a continuous-time financial market with Epstein-Zin-type recursive intertemporal preferences and performance concerns. We derive explicitly an equilibrium for the finite player and the mean-field version of the game, based on a study of geometric backward stochastic differential equations of Bernoulli type that describe the best replies of traders. Our results show that Epstein-Zin preferences can lead to substantially different equilibrium behavior.
Paper Structure (16 sections, 9 theorems, 144 equations)

This paper contains 16 sections, 9 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. N-player Games with Epstein-Zin Preferences
  4. Mean-Field Games with Epstein-Zin Preferences
  5. The Economics of Relative Performance Concerns with Recursive Preferences
  6. The Mean-Field Game as a Limit of Finite Player Games
  7. Preliminary results on geometric-Bernoulli BSDEs
  8. Geometric-Bernoulli BSDEs
  9. Solvability of geometric-Bernoulli BSDEs
  10. Optimizing against simple strategies
  11. Proof of the Main Theorems
  12. Proof of Theorem \ref{['thm existence NE']}
  13. Proof of Theorem \ref{['thmfg']}
  14. Proof of Theorem \ref{['thm convergence']}
  15. Proof of Theorem \ref{['theorem approximation']}
  16. ...and 1 more sections

Key Result

Theorem 2.2

There exists a unique Nash equilibrium in simple strategies $\hat{\boldsymbol{\alpha}}_N = (\hat{\alpha}_{1,N},..., \hat{\alpha}_{N,N})$ given by where

Theorems & Definitions (15)

  • Remark 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Remark 2.4
  • Definition 2.5
  • Theorem 2.6
  • Corollary 2.7
  • Theorem 2.8: Convergence to the MFGE
  • Theorem 2.9: Approximate NE
  • Lemma 3.1
  • ...and 5 more