主要目的是在局部 Lipschitz 条件下建立非线性 It{\^o} 随机微分方程的基本理论,包括解的存在性和非零性.过去文献中的局部 Lipschitz 条件被减弱为广义局部 Lipschitz 条件,其系数可以是局部、变系数、非线性的,在时间维上真正允许系数的时变性,在空间维上真正允许系数的非线性性.

The main purpose of this paper is to establish the fundamental theory of nonlinear It{\^o} stochastic differential equations under the Local Lipschitz condition, including the existence and non-zero property for the solutions. The local Lipschitz condition is weakened to the generalized local Lipschitz condition, which is local, variable and nonlinear, admits nearly arbitrary variability in the time and real nonlinearity in the state.