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

We consider the blow-up problems of the power type of stochastic differential equation, \begin{document}$ dX = \alpha X^p(t)dt+X^q(t)dW(t) $\end{document} . It has been known that there exists a critical exponent such that if \begin{document}$ p $\end{document} is greater than the critical exponent then the solution \begin{document}$ X(t) $\end{document} 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.