On the Global Positivity Solutions of Non-homogeneous Stochastic Differential Equations - Mhlanga, Farai Julius; Rundora, Lazarus (Frontiers in Applied Mathematics and Statistics, 2022)
In this article, we treat the existence and uniqueness of strong solutions to the Cauchy problem of stochastic equations of the form dXt=αXtdt+σXtγdBt,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.