Two fixed point theorems in complete random normed modules and their applications to backward stochastic equations

Abstract

This paper first proves two fixed point theorems in complete random normed modules, which are respectively the random generalizations of the classical Banach's contraction mapping principle and Browder--Kirk's fixed point theorem. As applications, the first is used to give the existence and uniqueness of solutions to various kinds of backward stochastic equations under $L^0$--Lipschitz assumptions and the second is used to establish the existence of solutions to backward stochastic equations of nonexpansive type.