pth moment asymptotic stability for neutral stochastic functional differential equations with Lévy processes

In this paper, we will consider a class of neutral stochastic functional differential equations with Lévy processes. Lévy processes contain a number of very important processes as special cases such as Brownian motion, the Poisson process, stable and self-decomposable processes and subordinators, and so on. But its sample paths are discontinuity, which makes the analysis more difficult. In this paper, we try to get over this difficulty. The contributions of this paper are as follows: (a) we will use Lyapunov functional method to study the pth moment asymptotic stability and almost sure asymptotic stability of neutral stochastic functional differential equations with Lévy processes; (b) under the result of (a), we will investigate two types of continuity of the solution: continuous in the pth moment and continuous in probability. Finally, we provide an example to illustrate the usefulness of the obtained results.