The Existence And The Stability Of Several Stochastic Differential Equations

For nearly half a century,the stochastic differential equation develop rapidly,at the same time,it has a wide range of applications in various fields.In this thesis,by using the theory of stochastic analysis,we study the existence and the stability properties of several stochastic differential equations.The thesis is organized as follows.For a stochastic prey-predator system with stage structure for the predator,we prove the existence of the unique global positive solution,and we give sufficient conditions for the global attractivity of the positive equilibrium.Based on the existence of the unique global positive solution of a stochastic cooperative system driven by white noise in a polluted environment,we get the asymptotical behavior of every species in the time average sense.For a class of stochastic Lotka-Voterra systems with infinite delay,we establish the conditions for the unique global positive solution.We introduce the concept of almost sure ?-extinction of every species in the system,and we give the sufficient conditions for the almost sure ?-extinction of every species in the system.Moreover,when the positive equilibrium point of the system exists,we prove that every non-zero solution is attracted by the positive equilibrium when the noise intensity is sufficiently small.For the stochastic fractional impulsive differential equations with infinite delay,we give the sufficient conditions for the existence of the mild solutions and the mean square stability of the solutions.An example is given to illustrate the effectiveness of our conclusions.We establish the concept of the square-mean S-asymptotically ?-periodic stochastic process.For a class of stochastic fractional differential equations driven by Lévy noise with piecewise arguments and a class of stochastic integral differential equations driven by Lévy noise with piecewise arguments,we prove the existence of the mild solutions of them and we give the sufficient conditions for the existence of the square-mean Sasymptotically ?-periodic solutions of them.Moreover,we show the global square-mean asymptotic stability of the square-mean S-asymptotically ?-periodic solutions for the stochastic integral differential equations driven by Lévy noise with piecewise arguments.We establish the concept of the S-asymptotically ?-periodic stochastic process in distribution.For a class of stochastic fractional functional differential equations driven by Lévy noise and a class of stochastic integral functional differential equations driven by Lévy noise,we prove the existence of the mild solutions of them and we give the sufficient conditions for the existence of the S-asymptotically ?-periodic solutions in distribution of them.Moreover,we show the global square-mean exponential asymptotic stability of the S-asymptotically ?-periodic solutions in distribution for the stochastic integral functional differential equations driven by Lévy noise.