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Infinite horizon McKean-Vlasov FBSDEs and applications to mean field control problems

Tianjiao Hua, Peng Luo

Abstract

In this paper, we study a class of infinite horizon fully coupled McKean-Vlasov forward-backward stochastic differential equations (FBSDEs). We propose a generalized monotonicity condition involving two flexible functions. Under this condition, we establish the well-posedness results for infinite horizon McKean-Vlasov FBSDEs by the method of continuation, including the unique solvability, an estimate of the solution, and the related continuous dependence property of the solution on the coefficients. Based on the solvability result, we study an infinite horizon mean field control problem. Moreover, by choosing appropriate form of the flexible functions, we can eliminate the different phenomenon between the linear-quadratic (LQ) problems on infinite horizon and finite horizon proposed in Wei and Yu (SIAM J. Control Optim. 59: 2594-2623, 2021).

Infinite horizon McKean-Vlasov FBSDEs and applications to mean field control problems

Abstract

In this paper, we study a class of infinite horizon fully coupled McKean-Vlasov forward-backward stochastic differential equations (FBSDEs). We propose a generalized monotonicity condition involving two flexible functions. Under this condition, we establish the well-posedness results for infinite horizon McKean-Vlasov FBSDEs by the method of continuation, including the unique solvability, an estimate of the solution, and the related continuous dependence property of the solution on the coefficients. Based on the solvability result, we study an infinite horizon mean field control problem. Moreover, by choosing appropriate form of the flexible functions, we can eliminate the different phenomenon between the linear-quadratic (LQ) problems on infinite horizon and finite horizon proposed in Wei and Yu (SIAM J. Control Optim. 59: 2594-2623, 2021).
Paper Structure (8 sections, 12 theorems, 136 equations)

This paper contains 8 sections, 12 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Infinite horizon McKean-Vlasov SDEs
  4. Infinite horizon McKean-Vlasov BSDEs
  5. Infinite horizon McKean-Vlasov FBSDEs
  6. Infinite horizon mean field control problems
  7. Pontryagin's stochastic maximum principle
  8. Solvability of mean field control problems

Key Result

Lemma 2.2

Let Assumption assumption x (i) (ii) hold. If the solution $X$ to McKean-Vlasov SDE mean field sde belongs to $L_{\mathbb{F}}^{2, K}\left(t, \infty ; \mathbb{R}^n\right)$, then we have

Theorems & Definitions (24)

  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • Definition 3.1
  • Remark 3.2
  • Theorem 3.3
  • Remark 3.4
  • ...and 14 more