# American Institute of Mathematical Sciences

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

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

 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.

Citation: Tetsuya Ishiwata, Young Chol Yang. Numerical and mathematical analysis of blow-up problems for a stochastic differential equation. Discrete and 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. [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. [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. [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. [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. [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. [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. [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. [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.

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##### 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. [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. [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. [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. [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. [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. [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. [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. [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.
Numerical solutions (4 samples)
Numerical Brownian motion
Histogram of $T_\tau^L$ and exact distribution of blow-up time (green)
Numerical solutions (3 samples)
Numerical Brownian motion
Distribution of numerical blow-up time
The number of non-blow-up solutions with fixed $T_{\max} = 1000$
The number of non-blow-up solutions with fixed $L = 1000$
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 $1000.101781$ $0.166273$ $0.00045478$
 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 $1000.101781$ $0.166273$ $0.00045478$
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
 $a$ 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
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
 $a$ 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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