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

Tetsuya Ishiwata; Young Chol Yang

doi:10.3934/dcdss.2020391

Article Contents

2021, Volume 14, Issue 3: 909-918. Doi: 10.3934/dcdss.2020391

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

Tetsuya Ishiwata and
Young Chol Yang^,

Shibaura Institute of Technology, 307 Fukasaku, Minuma, Saitama 337-8570, Japan

^* Corresponding author: Young Chol Yang
^* Corresponding author: Young Chol Yang

Received: December 31, 2018

Revised: February 29, 2020

Early access: June 2020

Published: March 2021

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

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Abstract

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.

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

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Figure 1. Numerical solutions (4 samples)

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Figure 2. Numerical Brownian motion

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

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Figure 4. Numerical solutions (3 samples)

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Figure 5. Numerical Brownian motion

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Figure 6. Distribution of numerical blow-up time

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

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

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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 $

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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

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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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References

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