Qualitative Analysis For Several Classes Of Stochastic And Impulsive Differential Equations

In this paper, we analyze the boundedness and stability of impulsive and stochas-tic differential equations with different types by means of the stochastic method, Lia-punov functions and fixed points method. The paper is divided into five parts.Partâ… mainly focus on describing the background, an overview of major workand notations needed in the later.In partâ…¡, we deal with the linear neutral differential equation with variable delaysand the linear neutral differential equation with variable delays and impulsesand their generalized forms. With fixed point method, we gives several sufficient con-ditions for the boundedness, asymptotic stability, exponential stability of the solution.Then, with fixed point theory and Borel-Contelli lemma, we established the sufficientconditions for the mean square boundedness, mean-square asymptotic stability, meansquare exponential stability and almost sure exponential stability sufficient conditionof the equationIn the Partâ…¢, we consider the impulsive differential equations on time scales by applying the Liapunov function method, we give a number of sufficient conditionsfor the boundedness and the exponential stability of the solutions. This part also con-tains the impulse method presented firstly to transform differential equations on timescales to the impulsive differential equations. Finally, we employe the impulse methodto study stochastic differential equations on time scalesand analyze the stability through comparing with that of equations without time scales.In partâ…£, the generalized stochastic Volterra equations with impulsive effect isconsidered.x(t0) = x0,Then, we present the sufficient conditions for the boundedness of the solution of theequation, as well as the exponential stability and the non-exponential stability by es-tablishing the new inequality on Liapunov function. The results is new.In the last partâ…¤, based on several models known, we propose a class of Volterra-type mean-reversion process to describe the bubble in security market. Three specialcases have been considered. For Caseâ… , we define the disappearance of the bubblecrash and estimate the probability. Caseâ…¡is described bywe demonstrate the non-negativity of the process firstly, and then employe the mar-tingale inequality to study the convergence of the process. Taking into account thechanges of the market states which may be described by the Markov chain, we alsoconsider the Markov modulated Volterra-type mean-reversion process as the third case by means of the limits to Brownian motion, we estimate Liapunov exponents in dif-ferent cases, and also gives the growth boundary in case of bounded nonlinear term.