The Study Of Well-posedness For Backward Stochastic Chaotic Systems Driven By Markov Chains And Poisson Jumps

Stochastic differential equations is an important branch of stochastic anal-ysis,since ito,the stochastic differential equations has been researched by many researchers and has achieved fruitful results.With the deepening of the research,the scholars put forward the concept of backward stochastic d-ifferential equations in the study of stochastic control problem,and do the further research.The backward stochastic differential equations not only has important theoretical value,but has a broad application prospect,such as backward stochastic differential equation to describe uncertain economic envi-ronment consumption preference and so on.In this paper,we mainly study the existence and uniqueness of the solutions for the backward stochastic Loren-z system driven by Markov chains and the backward unified chaotic system driven by Poisson jumps,the research contents can provide theoretical basis for stochastic control of chaotic systems under different noise disturbances.In the 1960s,E.N.Lorenz proposed the Lorenz model in the study of atmospheric convection:(?)of which a,b,c are parameters.Subsequently,many scholars have studied the properties of Lorenz system,such as the attractor,branch and other dynamical behaviors of stochastic Lorenz system.Lorenz system has aiso been used in physics,physiology,chaotic secure communication and other fields.This article will first study the existence and uniqueness of the solutions to backward stochastic Lorenz system.In 2008,Cohen and Elliott[l]studied the backward stochastic differential equation driven by Markov chains,and concluded that when the coefficient in the equation satisfies the Lipschitz con-dition,the backward stochastic differential equation driven by Markov chains has unique solution.However,unlike the literature[1],the backward stochas-tic Lorenz equation driven by Markov chains considered in this paper satisfies the local Lipschitz condition.In 2000,Lv and Chen studied the following unified chaotic system(see[17]):(?)of which the parameter ? ?[0,1].When the parameter ? changes,the famous Lorenz system,Chen system and Lv system can be obtained.The second part of this paper will study the backward stochastic unified chaotic system.In 2007,Sundar and Yin[3]studied the existence and unique-ness of the solutions for backward stochastic Lorenz system driven by Brownian motion.Due to the stochastic disturbances with Brownian motion are usually continuous,but not all the stochastic disturbances from outside are continuous,so it is necessary to consider the discontinuous stochastic disturbances.If the stochastic disturbance not only has continuous part,but also has discontinu-ous part,the solution for the backward chaotic systems whether exists and is unique?In this paper,the backward stochastic unified chaotic system driven by Poisson jumps will be considered,and the existence and uniqueness of the solutions will be investigated.This article is arranged as follows.This paper is divided into three chapters.The first chapter introduces the development of stochastic differential equations and some preliminary knowl-edge;In the second chapter,we discuss the existence and uniqueness of the solution for the backward stochastic Lorenz system driven by Markov chains;In the third chapter,we discuss the existence and uniqueness of the solution for the backward stochastic unified chaotic system driven by Poisson jumps.