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Mean-field backward stochastic differential equations and nonlocal PDEs with quadratic growth

Tao Hao, Ying Hu, Shanjian Tang, Jiaqiang Wen

Abstract

In this paper, we study general mean-field backward stochastic differential equations (BSDEs, for short) with quadratic growth. First, the existence and uniqueness of local and global solutions are proved with some new ideas for a one-dimensional mean-field BSDE when the generator $g\big(t, Y, Z, \mathbb{P}_{Y}, \mathbb{P}_{Z}\big)$ has a quadratic growth in $Z$ and the terminal value is bounded. Second, a comparison theorem for the general mean-field BSDEs is obtained with the Girsanov transform. Third, we prove the convergence of the particle systems to the mean-field BSDEs with quadratic growth, and the convergence rate is also given. Finally, in this framework, we use the mean-field BSDE to provide a probabilistic representation for the viscosity solution of a nonlocal partial differential equation (PDE, for short) as an extended nonlinear Feynman-Kac formula, which yields the existence and uniqueness of the solution to the PDE.

Mean-field backward stochastic differential equations and nonlocal PDEs with quadratic growth

Abstract

In this paper, we study general mean-field backward stochastic differential equations (BSDEs, for short) with quadratic growth. First, the existence and uniqueness of local and global solutions are proved with some new ideas for a one-dimensional mean-field BSDE when the generator has a quadratic growth in and the terminal value is bounded. Second, a comparison theorem for the general mean-field BSDEs is obtained with the Girsanov transform. Third, we prove the convergence of the particle systems to the mean-field BSDEs with quadratic growth, and the convergence rate is also given. Finally, in this framework, we use the mean-field BSDE to provide a probabilistic representation for the viscosity solution of a nonlocal partial differential equation (PDE, for short) as an extended nonlinear Feynman-Kac formula, which yields the existence and uniqueness of the solution to the PDE.
Paper Structure (4 sections, 4 theorems, 54 equations)

This paper contains 4 sections, 4 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Existence and Uniqueness
  4. Local solution

Key Result

Proposition 2.2

Let $p \in(1, \infty)$ and $M$ be a one-dimensional continuous $B M O$ martingale. If $\|M\|_{B M O(\mathbb{P})}<\Phi(p)$, then $\mathscr{E}(M)$ satisfies the reverse Hölder inequality: for any stopping time $\tau$, with a positive constant $c_p$ depending only on $p$.

Theorems & Definitions (8)

  • Definition 2.1
  • Proposition 2.2: The Reverse Hölder Inequality
  • Proposition 2.3
  • Remark 3.1
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • proof