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=8

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=<p>In this article, we treat the existence and uniqueness of strong solutions to the Cauchy problem of stochastic equations of the form <inline-formula><mml:math id="M1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>d</mml:mtext><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>α</mml:mi><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mtext>d</mml:mtext><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mi>σ</mml:mi><mml:msubsup><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>γ</mml:mi></mml:mrow></mml:msubsup><mml:mtext>d</mml:mtext><mml:msub><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>x</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo></mml:math></inline-formula> 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 &gt; 1 are considered. A complete description of every possible behavior of the process <italic>X</italic><sub><italic>t</italic></sub> at the boundary points of the state interval is provided. For applications, the Cox-Ingersoll-Ross model is considered.</p>