The Stability Of One-Dimensional Stochastic Differential Equations

Stochastic differential equations have a very wide range of applications in many areas, such as economics, population ecology and so on. Given an ordinary differential equation, its solution may be unstable, while the corresponding stochastic differential equation might be stable.This paper studies the existence and uniqueness of the solutions, boundedness and stability issues of the stochastic differential equations.The first chapter introduces the background of this work and the main work of this paper. The second chapter describes the properties of the solutions of the stochastic differential equations dx(t)= f(x(t),t)dt+g(x(t),t)dB(t), when the coefficient/and g of the stochastic differential equations satisfy the Lipschitz conditions and linear growth conditions, including the existence and uniqueness of the solutions, stability in probabil-ity, almost sure exponential stability and moment exponential stability and so on. In the last two sections of the second chapter, we have studied for a given stochastic differential system. On one hand, if the solution is for the exponential growth,then the noise can turn it into a new system, its solution is for the polynomial growth. On the other hand, if its solution is bounded, the noise can also turn it into a new system,its solution is for the exponential growth. All in all, the noise can not only promote the exponential growth of stochastic differential equations but also suppression the exponential growth of the equations. In the third chapter, we have studied the stability of the stochastic differential equations dx(t)= f(x(t),t)dt+g(x(t),t)dB(t),hoth of the coefficients/and g satisfy Lipschitz conditions f satisfies the unilateral polynomial growth condition, and g satisfy the polynomial growth condition. This chapter will explain that appro-priateβcan guarantee the existence and uniqueness of the global solution of stochastic differential equations,and there is a constant Kp dependent only on the initial value, such that the solution of the equations are bounded.Finally, we will also discuss the q large enough to ensure that the solution of the system to meet the almost sure exponential stability. Finally, corresponding examples are given for demonstration.