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Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow
Numerical and mathematical analysis of blow-up problems for a stochastic differential equation
Shibaura Institute of Technology, 307 Fukasaku, Minuma, Saitama 337-8570, Japan |
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.
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. |
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. |
[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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Sample No. | |||
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0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | |
977 | 941 | 888 | 791 | 665 | 486 | 369 | 226 | 118 | |
995 | 988 | 968 | 880 | 756 | 566 | 377 | 223 | 122 |
0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | |
977 | 941 | 888 | 791 | 665 | 486 | 369 | 226 | 118 | |
995 | 988 | 968 | 880 | 756 | 566 | 377 | 223 | 122 |
0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | |
995 | 988 | 968 | 880 | 756 | 566 | 377 | 223 | 122 | |
996 | 989 | 940 | 848 | 668 | 401 | 234 | 105 | 44 |
0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | |
995 | 988 | 968 | 880 | 756 | 566 | 377 | 223 | 122 | |
996 | 989 | 940 | 848 | 668 | 401 | 234 | 105 | 44 |
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