## Annals of Probability

- Ann. Probab.
- Volume 37, Number 4 (2009), 1524-1565.

### Mean-field backward stochastic differential equations: A limit approach

Rainer Buckdahn, Boualem Djehiche, Juan Li, and Shige Peng

#### Abstract

Mathematical mean-field approaches play an important role in different fields of Physics and Chemistry, but have found in recent works also their application in Economics, Finance and Game Theory. The objective of our paper is to investigate a special mean-field problem in a purely stochastic approach: for the solution (*Y*, *Z*) of a mean-field backward stochastic differential equation driven by a forward stochastic differential of McKean–Vlasov type with solution *X* we study a special approximation by the solution (*X*^{N}, *Y*^{N}, *Z*^{N}) of some decoupled forward–backward equation which coefficients are governed by *N* independent copies of (*X*^{N}, *Y*^{N}, *Z*^{N}). We show that the convergence speed of this approximation is of order $1/\sqrt{N}$. Moreover, our special choice of the approximation allows to characterize the limit behavior of $\sqrt{N}(X^{N}-X,Y^{N}-Y,Z^{N}-Z)$. We prove that this triplet converges in law to the solution of some forward–backward stochastic differential equation of mean-field type, which is not only governed by a Brownian motion but also by an independent Gaussian field.

#### Article information

**Source**

Ann. Probab., Volume 37, Number 4 (2009), 1524-1565.

**Dates**

First available in Project Euclid: 21 July 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1248182147

**Digital Object Identifier**

doi:10.1214/08-AOP442

**Mathematical Reviews number (MathSciNet)**

MR2546754

**Zentralblatt MATH identifier**

1176.60042

**Subjects**

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]

Secondary: 60B10: Convergence of probability measures

**Keywords**

Backward stochastic differential equation mean-field approach McKean–Vlasov equation mean-field BSDE tightness weak convergence

#### Citation

Buckdahn, Rainer; Djehiche, Boualem; Li, Juan; Peng, Shige. Mean-field backward stochastic differential equations: A limit approach. Ann. Probab. 37 (2009), no. 4, 1524--1565. doi:10.1214/08-AOP442. https://projecteuclid.org/euclid.aop/1248182147