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 \[dX_t=α X_t dt+σ X_t^γ dB_t, X_0=x>0.\] 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 γ=2, γ=12 and the general case, that is, γ>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.