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March  2021, 14(3): 909-918. doi: 10.3934/dcdss.2020391

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

Received  January 2019 Revised  March 2020 Published  March 2021 Early access  June 2020

Fund Project: 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.

Keywords: SDE, blow-up, blow-up time, Euler-Maruyama scheme, adaptive time step control.
Mathematics Subject Classification: Primary: 60H10, 65C20; Secondary: 34F05.
Citation: Tetsuya Ishiwata, Young Chol Yang. Numerical and mathematical analysis of blow-up problems for a stochastic differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 909-918. doi: 10.3934/dcdss.2020391
References:
[1]

J. A. D. Appleby, G. Berkolaiko and A. Rodkina, Non-exponential stability and decay rates in nonlinear stochastic differential equations with unbounded noise, Stochastics, 81 (2009), 99-127. doi: 10.1080/17442500802088541.  Google Scholar

[2]

J. A. D. Appleby, X. R. Mao and A. Rodkina, Stabilization and destabilization of nonlinear differential equations by noise, IEEE Trans. Automat. Control, 53 (2008), 683-691.  doi: 10.1109/TAC.2008.919255.  Google Scholar

[3]

J. Davila, J. F. Bonder, J. D. Rossi, P. Groisman and M. Sued, Numerical analysis of Stochastic differential equations with explosions, Stochastic Analysis and Applications, 23 (2005), 809-825.  doi: 10.1081/SAP-200064484.  Google Scholar

[4]

P. Groisman and J. D. Rossi., Explosion time in stochastic differential equations with small diffusion, Electoric Journal of Differential Equations, 2007 (2007), 9 pp.  Google Scholar

[5]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Second edition, Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0949-2.  Google Scholar

[6]

C. Kelly, A. Rodkina and E. M. Rapoo, Adaptive timestepping for pathwise stability and positivity of strongly discretised nonlinear stochastic differential equations, J. Comput. Appl. Math., 334 (2018), 39-57.  doi: 10.1016/j.cam.2017.11.027.  Google Scholar

[7]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Applications of Mathematics (New York), 23. Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.  Google Scholar

[8]

J. E. Macías-Díaz and J. Villa-Morales, Finite-difference modeling à la Mickens of the distribution of the stopping time in a stochastic differential equation, Journal of Difference Equations and Applications, 23 (2017), 799-820.  doi: 10.1080/10236198.2017.1284828.  Google Scholar

[9]

J. M. Sanz-Serna and J. G. Verwer, A study of the recursion $y_{n+1}=y_n+\tau{y_n^m}$, J. Math. Anal. Appl., 116 (1986), 456-464.  doi: 10.1016/S0022-247X(86)80010-5.  Google Scholar

show all references

References:
[1]

J. A. D. Appleby, G. Berkolaiko and A. Rodkina, Non-exponential stability and decay rates in nonlinear stochastic differential equations with unbounded noise, Stochastics, 81 (2009), 99-127. doi: 10.1080/17442500802088541.  Google Scholar

[2]

J. A. D. Appleby, X. R. Mao and A. Rodkina, Stabilization and destabilization of nonlinear differential equations by noise, IEEE Trans. Automat. Control, 53 (2008), 683-691.  doi: 10.1109/TAC.2008.919255.  Google Scholar

[3]

J. Davila, J. F. Bonder, J. D. Rossi, P. Groisman and M. Sued, Numerical analysis of Stochastic differential equations with explosions, Stochastic Analysis and Applications, 23 (2005), 809-825.  doi: 10.1081/SAP-200064484.  Google Scholar

[4]

P. Groisman and J. D. Rossi., Explosion time in stochastic differential equations with small diffusion, Electoric Journal of Differential Equations, 2007 (2007), 9 pp.  Google Scholar

[5]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Second edition, Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0949-2.  Google Scholar

[6]

C. Kelly, A. Rodkina and E. M. Rapoo, Adaptive timestepping for pathwise stability and positivity of strongly discretised nonlinear stochastic differential equations, J. Comput. Appl. Math., 334 (2018), 39-57.  doi: 10.1016/j.cam.2017.11.027.  Google Scholar

[7]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Applications of Mathematics (New York), 23. Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.  Google Scholar

[8]

J. E. Macías-Díaz and J. Villa-Morales, Finite-difference modeling à la Mickens of the distribution of the stopping time in a stochastic differential equation, Journal of Difference Equations and Applications, 23 (2017), 799-820.  doi: 10.1080/10236198.2017.1284828.  Google Scholar

[9]

J. M. Sanz-Serna and J. G. Verwer, A study of the recursion $y_{n+1}=y_n+\tau{y_n^m}$, J. Math. Anal. Appl., 116 (1986), 456-464.  doi: 10.1016/S0022-247X(86)80010-5.  Google Scholar

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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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
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$ 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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$ 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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