Qualitative Researches Of Random Impulsive Differential Equations

It is well-known that the theory of stochastic impulsive differential equations not only is richer than ordinary impulsive differential equations but also represents a more practical and exact framework for mathematical modeling of many real phenomena. These years, significant progress has been made in the theories including existence, uniqueness, bound-edness and stability. However, all of these are just the beginning of the theories, especially the corresponding qualitative theory is still at the stage of its development. In this thesis, we establish models of stochastic impulsive differential equations and give some far more practical results by weakening the conditions of known theories. From the financial engineering point of view, the results are full of practical meanings and can be used to provide reliable decision-making for our work. We mainly study the existence, uniqueness, p-moment boundedness and stability of stochastic impulsive differential equations in this thesis.In Chapter 2, we study the existence and uniqueness of stochastic impulsive differential equations by Pearson iteration method. The technique employed to achieve the desired results includes the Cauchy-Schwarz inequality, Lipschtiz conditions, and some techniques in stochastic analysis. This chapter is the basis of other theories.In Chapter 3, we do researches on the boundedness of stochastic impulsive differential equations. By means of Liapunov's indirect method, some new sufficient conditions for uniform(uniform and ultimate, uniform and uniformly ultimate) p-moment boundedness of these systems are presented, where (dV(t,x(t))/(dt))is imposed on only a little restriction even to obtain uniform boundedness and uniformly ultimate boundedness. The obtained results greatly improve their corresponding known results. The method can be used to apply in other fields research of stochastic impulsive differential equations.In Chapter 4, we present the results of the stability of disturbed stochastic impulsive differential equations. Firstly, for simple n-dimention system of stochastic impulsive differential equations, we give some sufficient conditions for stability in mean square. Secondly, by means of comparison methods, we obtain sufficient conditions for some simple disturbed systems. At last, with the conclusions above, we get the results for general system of disturbed stochastic impulsive differential equations.