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 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.