Mean-field backward stochastic differential equations and applications

TL;DR

This paper investigates mean-field backward stochastic differential equations, proving existence, uniqueness, and a comparison theorem for a subclass, with applications to finance utility optimization.

Contribution

It establishes foundational results for mean-field bsde, including existence, uniqueness, and a comparison theorem, and provides explicit solutions for linear cases with financial applications.

Findings

Proved existence and uniqueness of solutions under mild conditions.

Established a comparison theorem for a subclass of mean-field bsde.

Derived explicit formulas for linear mean-field bsde solutions.

Abstract

In this paper we study the mean-field backward stochastic differential equations (mean-field bsde) of the form dY(t) =-f(t,Y(t),Z(t),K(t, . ),E[\varphi(Y(t),Z(t),K(t,.))])dt+Z(t)dB(t) +\int_{R_{0}}K(t,\zeta)\tilde{N}(dt,d\zeta), where B is a Brownian motion, \tilde{N} is the compensated Poisson random measure. Under some mild conditions, we prove the existence and uniqueness of the solution triplet (Y,Z,K). It is commonly believed that there is no comparison theorem for general mean-field bsde. However, we prove a comparison theorem for a subclass of these equations. When the mean-field bsde is linear, we give an explicit formula for the first component Y(t) of the solution triplet. Our results are applied to solve a mean-field recursive utility optimization problem in finance.