Fundamental Properties of Nonlinear Stochastic Differential Equations

Abstract

The existence of solutions is used the premise of discussing other properties of dynamic systems. The goal of this paper is to investigate the fundamental properties of nonlinear stochastic differential equations via the Khasminskii test, including the local existence and global existence of the solutions. Firstly, a fundamental result is given as a lemma to verify the local existence of solutions to the considered equation. Then, the equivalent proposition for the global existence and the fundamental principle for the Khasminskii test are formally established. Moreover, the classical Khasminskii test is generalized to the cases with high-order estimates and heavy nonlinearity for the stochastic derivatives of the Lyapunov functions. The role of the noise in this aspect is especially investigated, some concrete criteria are obtained, and an application for the role of the noise in the persistence of financial systems is accordingly provided. As another application of the fundamental principle, a new version of the Khasminskii test is established for the delayed stochastic systems. Finally the conclusions obtained in the paper are verified by simulation. The results show that, under weaker conditions, the global existence of better solutions to stochastic systems to those in the existing literature can be obtained.