Existence, Uniqueness And Stability For Some Stochastic Differential Equations

Stochastic differential equation is not only an important cognitive tool of scientific theory,but also it plays an indispensable role in real life. Hence, it is necessary to study the theories ofstochastic differential equations. Among these theories, the existence, uniqueness and stabilityof solutions to stochastic differential equations occupy a significant position.The paper contains five chapters, and we mainly study the existence, uniqueness andstability of several stochastic differential equations.In Chapter1, we first introduce the importance of stochastic differential equations. We alsomention the research methods and development status of stochastic differential equations.Besides, we present some preliminary knowledge of this paper.In Chapter2, we establish that the solution of stochastic Volterra-Levin equations isexistent, unique and stable in mean square based on the successive approximations, It isometryand H lder inequality. Moreover, two examples are provided to illustrate the effectiveness ofthe obtained results.In Chapter3, by constructing proper Banach space and contraction mapping and using theBanach fixed point theory, we study the existence, uniqueness and p th moment exponentialstability of the solution of stochastic Volterra-Levin equations with Poissin jumps. Then, basedon the Borel-Cantelli lemma, we prove the solution is almost sure p th moment exponentialstable. Finally, through examples and comparisons, we get that our results improve andgeneralize those given in the previous literature.In Chapter4, we mainly apply the Banach fixed point theory, H lder inequality andBukh lder-Davis-Gundy inequality to study the existence, uniqueness and p th momentasymptotical stability of stochastic neutral differential equations with delays and Poisson jumps.By compare with the previous literature, we find our results are more general. As an application,we finally give an example to illustrate our results.In Chapter5, we summarize the main work and innovation in the paper, and propose thedirection of the improvement of this paper.