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July 2009 Mean-field backward stochastic differential equations: A limit approach
Rainer Buckdahn, Boualem Djehiche, Juan Li, Shige Peng
Ann. Probab. 37(4): 1524-1565 (July 2009). DOI: 10.1214/08-AOP442


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 (XN, YN, ZN) of some decoupled forward–backward equation which coefficients are governed by N independent copies of (XN, YN, ZN). 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.


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Rainer Buckdahn. Boualem Djehiche. Juan Li. Shige Peng. "Mean-field backward stochastic differential equations: A limit approach." Ann. Probab. 37 (4) 1524 - 1565, July 2009.


Published: July 2009
First available in Project Euclid: 21 July 2009

zbMATH: 1176.60042
MathSciNet: MR2546754
Digital Object Identifier: 10.1214/08-AOP442

Primary: 60H10
Secondary: 60B10

Keywords: backward stochastic differential equation , McKean–Vlasov equation , mean-field approach , mean-field BSDE , tightness , weak convergence

Rights: Copyright © 2009 Institute of Mathematical Statistics


Vol.37 • No. 4 • July 2009
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