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

Abstract

We consider the blow-up problems of the power type of stochastic differential equation, dX = αX 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.