On the Global Positivity Solutions of Non-homogeneous Stochastic Differential Equations

In this article, we treat the existence and uniqueness of strong solutions to the Cauchy problem of stochastic equations of the form dXt=αXtdt+σXγtdBt,X0=x>0. The construction does not require the drift and the diffusion coefficients to be Lipschitz continuous. Sufficient and necessary conditions for the existence of a global positive solution of non-homogeneous stochastic differential equations with a non-Lipschitzian diffusion coefficient are sought using probabilistic arguments. The special case γ = 2 and the general case, that is, γ > 1 are considered. A complete description of every possible behavior of the process Xt at the boundary points of the state interval is provided. For applications, the Cox-Ingersoll-Ross model is considered.