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

TL;DR

This paper extends classical fixed point theorems to complete random normed modules and applies these results to establish existence and uniqueness of solutions for various backward stochastic equations.

Contribution

It introduces two fixed point theorems in complete random normed modules, generalizing classical results and applying them to backward stochastic equations.

Findings

Existence and uniqueness of solutions to backward stochastic equations under $L^0$--Lipschitz conditions

Solutions to backward stochastic equations of nonexpansive type established

Fixed point theorems generalized to random normed modules

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.