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Regularity of the Value Function in Discounted Infinite-Time Mean Field Games

Yongsheng Song, Zeyu Yang

Abstract

In [17], we introduced the discounted infinite-time mean field games. Subsequently, in [18], we studied the connection between infinite-time mean field FBSDEs and elliptic master equations. In this paper, we further investigate the regularity of the representative player's value function. Specifically, we first prove the strong existence and uniqueness, as well as the uniqueness in law, for an extended class of infinite-time FBSDEs. We then establish the Lions-differentiability for the derivative of the representative player's value function with respect to the measure argument, and provide an explicit characterization for it using solutions to FBSDEs.

Regularity of the Value Function in Discounted Infinite-Time Mean Field Games

Abstract

In [17], we introduced the discounted infinite-time mean field games. Subsequently, in [18], we studied the connection between infinite-time mean field FBSDEs and elliptic master equations. In this paper, we further investigate the regularity of the representative player's value function. Specifically, we first prove the strong existence and uniqueness, as well as the uniqueness in law, for an extended class of infinite-time FBSDEs. We then establish the Lions-differentiability for the derivative of the representative player's value function with respect to the measure argument, and provide an explicit characterization for it using solutions to FBSDEs.
Paper Structure (8 sections, 9 theorems, 123 equations)

This paper contains 8 sections, 9 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Solutions to infinite-time FBSDEs
  4. Strong solutions to infinite-time FBSDEs
  5. Distributional uniqueness
  6. Lions-differentiability of the value function's derivative
  7. Existence of directional derivative
  8. Representation of the Lions-derivative

Key Result

Theorem 3.2

Under Assumption assum: fbsde, for each ${\cal F}_0$-measurable square integrable random variable $\xi$ , (eq: fbsde) has a unique solution $(X_t,Y_t,Z_t)$ in $L_K^2$.

Theorems & Definitions (12)

  • Theorem 3.2
  • Definition 3.3: Strong uniqueness
  • Definition 3.4: Weak uniqueness
  • Theorem 3.5
  • Lemma 4.2
  • Corollary 4.3
  • Theorem 4.4
  • Theorem 4.5
  • Lemma 4.6
  • Remark 4.7
  • ...and 2 more