The Stability Of Some Kinds Of Stochastic Delay Differential Equations And Stochastic Volterra Integral Equations

In the past several decades, stochastic delay differential equations and stochasticVolterra integral equations have been widely applied in many fields of science, such as inautomatic control, biology, chemical reaction engineering, medicine, economics, demog-raphy etc. Since these equations can only be solved explicitly in some special cases, it isnecessary to develop the qualitative theory, in which stability theory plays an importantrole. In this dissertation, it is studied the stability of some kinds of stochastic delay dif-ferential equations and stochastic Volterra integral equations. This dissertation is dividedinto five chapters as follows:In the first chapter, it is reviewed brie?y the developing history and some results onthe theory of stability of stochastic differential equations. And some notations, definitionsand useful lemmas are introduced.In Chapter 2, the stochastic stability for the stochastic functional differential equa-tions with finite delay is analyzed. It is well-known that Liapunov functional method is animportant method to study the stability of functional differential equations. However, itcan not be applied inconveniently for the stochastic functional differential equations. Toget rid of the difficulty, a new method, Liapunov quasi-functional method, is proposed.Making use of this method it is derived the sufficient conditions, respectively, for thestochastic stability, the stochastic asymptotical stability and the stochastic global asymp-totical stability of the stochastic functional differential equations with finite delay.In Chapter 3, for a class of stochastic functional differential equations with an infi-nite delay, i.e. stochastic Volterra integro- differential equations with infinite delay, theconditions of the stochastic stability, the stochastic asymptotical stability and the stochas-tic global asymptotical stability are obtained by the Liapunov quasi-functional method.The results show that some Volterra integro-differential equations with infinite delay arestochastic stable under a small random perturbation. Meanwhile, it is derived the con-ditions of the intensity of the random perturbation, under which the perturbed equationshave better stability, such as stochastic asymptotical stability and stochastic global asymp-totical stability. In Chapter 4, a class of stochastic functional differential equations with a variabledelay, i.e. stochastic Volterra integro-differential equations, are considered. By the Li-apunov quasi-functional method, the conditions of the stochastic stability, the stochasticasymptotical stability and the stochastic global asymptotical stability are given. In thesimilar results as in Chapter 3, it is obtained the conditions of the intensity of the randomperturbation, under which the perturbed equations have better stability.In Chapter 5, the stochastic stability and the moment exponential stability of somestochastic Volterra integral equations are analyzed. Having introduced the concept ofquasi-Ito? process, the Ito? formula is extended to a more general form, such that it couldbe applied to study the stochastic Volterra integral equations. And then the existence anduniqueness theorem of the solution to these equations are proved by using the contractive-mapping principle. The sufficient conditions of the stochastic stability, the stochasticasymptotical stability, the stochastic global asymptotical stability and the moment expo-nential stability are obtained by applied the generalized Ito? formula.Moreover, after the stability theorem in every chapter, some examples are given toshow that the results are applicable. At the end of the dissertation, the conclusions, theinnovation and our future work are summarized.