Existence-uniqueness for Stochastic Functional Differential Equations with Non-Lipschitz Coefficients

The main aim of this paper is to develop some basic theories of stochastic functional differential equations (SFDEs) with non-Lipschitz coefficients.Firstly, we show that Peano's theorem holds for SFDEs, that is, the continuity alone is sufficient to prove the local existence of the initial value problem of SFDEs.Secondly, some new uniqueness theorems are established by the comparison methods proposed by Xu et al.And then, continuation theorems and global existence theorems for SFDEs with non-Lipschitz coefficients are obtained.Finally, an example is given to illustrate the efficiency of the obtained results.