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Continuous-Time Heterogeneous Agent Models with Recursive Utility and Preference for Late Resolution

Yves Achdou, Qing Tang

TL;DR

It is proved the existence and uniqueness of a constraint viscosity solution to the Hamilton-Jacobi-Bellman equation, which gives existence to the value function of the optimal control problem.

Abstract

We consider continuous-time heterogeneous agent models with recursive utility cast as mean field games, in which agents prefer late resolution of uncertainty. We prove the existence and uniqueness of a constraint viscosity solution to the Hamilton-Jacobi-Bellman equation, which gives existence to the value function of the optimal control problem. We investigate the existence of solutions to the mean field game system and discuss some important qualitative features of the model.

Continuous-Time Heterogeneous Agent Models with Recursive Utility and Preference for Late Resolution

TL;DR

It is proved the existence and uniqueness of a constraint viscosity solution to the Hamilton-Jacobi-Bellman equation, which gives existence to the value function of the optimal control problem.

Abstract

We consider continuous-time heterogeneous agent models with recursive utility cast as mean field games, in which agents prefer late resolution of uncertainty. We prove the existence and uniqueness of a constraint viscosity solution to the Hamilton-Jacobi-Bellman equation, which gives existence to the value function of the optimal control problem. We investigate the existence of solutions to the mean field game system and discuss some important qualitative features of the model.
Paper Structure (12 sections, 33 theorems, 229 equations, 6 figures, 1 table)

This paper contains 12 sections, 33 theorems, 229 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Analysis of the Hamilton-Jacobi-Bellman equation
  4. Analysis of the Fokker-Planck-Kolmogorov equation
  5. Existence of solution to the Mean Field Game system
  6. Nonexistence of invariant measures when $r=\rho$
  7. Existence of solutions to the MFG system
  8. Numerical examples
  9. Discussion and future works
  10. Proof of the strong comparison principle
  11. Asymptotic analysis as $x\to +\infty$ in the case $r=\rho$
  12. Asymptotic analysis near $x={\underline x}$

Key Result

Lemma 2.1

For $\rho>r$, the parameter $b$ introduced in tab1 is such that $r<b<\rho$.

Figures (6)

  • Figure 1: Consumption with $\gamma=2$: Test 1 and Test 2
  • Figure 2: Consumption with $\psi=0.4$: Test 3 and Test 4
  • Figure 3: Saving with $\gamma=2$: Test 1 and Test 2
  • Figure 4: Saving with $\psi=0.4$: Test 3 and Test 4
  • Figure 5: Asset distribution with $\gamma=2$: Test 1 and Test 2
  • ...and 1 more figures

Theorems & Definitions (68)

  • Lemma 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • Lemma 3.4
  • proof
  • ...and 58 more