Further results on existence-uniqueness for stochastic functional differential equations

Abstract

The aim of this paper is to develop some basic theories of stochastic functional differential equations (SFDEs) under the local Lipschitz condition in continuous functions space C. Firstly, we establish a global existence-uniqueness lemma for the SFDEs under the global Lipschitz condition in C without the linear growth condition. Then, under the local Lipschitz condition in C, we show that the non-continuable solution of SFDEs still exists if the drift coefficient and diffusion coefficient are square-integrable with respect to t when the state variable equals zero. And the solution of the considered equation must either explode at the end of the maximum existing interval or exist globally. Furthermore, some more general sufficient conditions for the global existenceuniqueness are obtained. Our conditions obtained in this paper are much weaker than some existing results. For example, we need neither the linear growth condition nor the continuous condition on the time t . Two examples are provided to show the effectiveness of the theoretical results.