Stochastic Differential Utility Based On Mean-Field Backward Stochastic Differential Equations

The conception of backward stochastic differential equations(BSDEs) was brought out to solve the terminal value problems. Nowadays, the BSDEs have achieved sufficient development and are widely used in mathematics, economic, financial and many other fields. In2007, Lasry and Lions derived the structure of mean-field backward stochastic differential equations from the limit equation of high-dimensional BSDEs. In economics, stochastic differential utility(SDU) refers to the satifaction of consumers on the current consumption processes. It was defined as the adapted solution of some BSDE by Duffie and Epstein.This article is mainly discussed the SDU based on mean-field BSDEs on the basis of previous researchs. Firstly, some basic knowledge and important theorems in BSDEs and mean-field BSDEs are given. Secondly, the SDU is redefined based on the mean-field BSDEs. In this article, the existence of SDU is proved by Picard ineration and the uniqueness and the comparison theorems are proved by Ito formula. Finally, some classic properties of SDU are discussed as well. Those properties include the continuity of utility function, the monotonicity of SDU, the concavity and andrist aversion of SDU and so on.The main conclusions of this article are important to promote the situations of classic backward stochastic differential equations. Therefore, this article has great theoretical and practical significance.