Article Contents
Article Contents

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

• * Corresponding author: Young Chol Yang

The first author is partly supported by JSPS KAKENHI Grant number 15H03632 and 19H05599

• 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.

Mathematics Subject Classification: Primary: 60H10, 65C20; Secondary: 34F05.

 Citation:

• Figure 1.  Numerical solutions (4 samples)

Figure 2.  Numerical Brownian motion

Figure 3.  Histogram of $T_\tau^L$ and exact distribution of blow-up time (green)

Figure 4.  Numerical solutions (3 samples)

Figure 5.  Numerical Brownian motion

Figure 6.  Distribution of numerical blow-up time

Figure 7.  The number of non-blow-up solutions with fixed $T_{\max} = 1000$

Figure 8.  The number of non-blow-up solutions with fixed $L = 1000$

Table 1.  Numerical paramaters at numerical blow-up time

 Sample No. $X_n$ $T_\tau^L$ $|W_n-M|$ 1 $1000.343314$ $0.043027$ $0.00076978$ 2 $1000.154401$ $0.39964$ $0.0007528$ 3 $1000.61063$ $0.0209$ $0.00040622$ 4 $1000.101781$ $0.166273$ $0.00045478$

Table 2.  The number of Blow-up solutions with fixed $T_{max} = 1000$

 $a$ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 $L=100$ 977 941 888 791 665 486 369 226 118 $L=1000$ 995 988 968 880 756 566 377 223 122

Table 3.  The number of non-blow-up solutions with fixed $L = 1000$

 $a$ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 $T_{\max}=100$ 995 988 968 880 756 566 377 223 122 $T_{\max}=1000$ 996 989 940 848 668 401 234 105 44
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Figures(8)

Tables(3)