AUTHOR=Mhlanga Farai Julius , Rundora Lazarus 

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

JOURNAL=Frontiers in Applied Mathematics and Statistics

VOLUME=Volume 8 - 2022

YEAR=2022

URL=https://www.frontiersin.org/journals/applied-mathematics-and-statistics/articles/10.3389/fams.2022.847896

DOI=10.3389/fams.2022.847896

ISSN=2297-4687

ABSTRACT=In this paper, we treat the existence and uniqueness of strong solutions to the Cauchy problem of stochastic equations of the form
\[\mathrm{d}X_t=\alpha X_t\,\mathrm{d}t+\sigma X_t^{\gamma}\,\mathrm{d}B_t,~~~X_0=x>0.\]
 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 cases $\gamma=2$, $\gamma=\frac{1}{2}$ and the general case, that is, $\gamma>1$ are considered. A complete description of every possible behavior of the process $X_t$ at the boundary points of the state interval is provided. For applications, the Cox-Ingersoll-Ross model is considered.