Mean-field backward stochastic differential equations: A limit approach

TL;DR

This paper investigates a limit approach to mean-field backward stochastic differential equations, demonstrating convergence of an approximation scheme and characterizing the limit behavior involving Gaussian fields.

Contribution

It introduces a novel approximation method for mean-field backward SDEs and establishes convergence rates and limit laws involving Gaussian fields.

Findings

Convergence speed of approximation is of order 1/√N.

The triplet (X^N, Y^N, Z^N) converges in law to a mean-field SDE solution.

Limit behavior involves an independent Gaussian field.

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.