Existence and Uniqueness of Solutions to Neutral Stochastic Functional Differential Equations with Poisson Jumps

Abstract

A class of neutral stochastic functional differential equations with Poisson jumps (NSFDEwPJs), $\text{d}[x(t)-G(x_t)]=f(x_t,t)\text{d}t+g(x_t,t)\text{d}W(t)+h(x_t,t)\text{d}N(t),t\in [t_0,T]$, with initial value $x_{t_0}=\xi =\{\xi (\theta ):-\tau \le \theta \le 0\}$, is investigated. First, we consider the existence and uniqueness of solutions to NSFDEwPJs under the uniform Lipschitz condition, the linear growth condition, and the contractive mapping. Then, the uniform Lipschitz condition is replaced by the local Lipschitz condition, and the existence and uniqueness theorem for NSFDEwPJs is also derived.