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

Abstract

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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.

Keywords

• geometric Brownian motion
• Itô diffusion
• Lipschitz continuous
• scale function
• speed measure