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

中文翻译：

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随机微分方程爆炸问题的数值和数学分析

我们考虑随机微分方程幂类型的爆破问题，即dX = \ alpha X ^ p（t）dt + X ^ q（t）dW（t）$。已知存在一个临界指数，使得如果$ p $大于临界指数，则解$ X（t）$几乎肯定会在有限时间内爆炸。在我们的研究中，针对这一关键指数，我们通过自适应时间步长提出了一种数值方案，并对其进行了数学分析。最后，我们使用所提出的方案显示了数值结果。