Numerical and mathematical analysis of blow-up problems for a stochastic differential equation

Abstract

<p style='text-indent:20px;'>We consider the blow-up problems of the power type of stochastic differential equation, <inline-formula><tex-math id="M1">\begin{document}$ dX = \alpha X^p(t)dt+X^q(t)dW(t) $\end{document}</tex-math></inline-formula>. It has been known that there exists a critical exponent such that if <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula> is greater than the critical exponent then the solution <inline-formula><tex-math id="M3">\begin{document}$ X(t) $\end{document}</tex-math></inline-formula> blows up almost surely in the finite time. In our research, focus on this critical exponent, we propose a numerical scheme by adaptive time step and analyze it mathematically. Finally we show the numerical result by using the proposed scheme.