
-
Previous Article
A mathematical analysis of an activator-inhibitor Rho GTPase model
- JCD Home
- This Issue
-
Next Article
Piecewise discretization of monodromy operators of delay equations on adapted meshes
Numerical preservation issues in stochastic dynamical systems by $ \vartheta $-methods
Department of Information Engineering and Computer Science and Mathematics, University of L'Aquila, Via Vetoio, Loc. Coppito - 67100 L'Aquila, Italy |
This paper analyzes conservation issues in the discretization of certain stochastic dynamical systems by means of stochastic $ \vartheta $-mehods. The analysis also takes into account the effects of the estimation of the expected values by means of Monte Carlo simulations. The theoretical analysis is supported by a numerical evidence on a given stochastic oscillator, inspired by the Duffing oscillator.
References:
[1] |
G. Berkolaiko, E. Buckwar, C. Kelly and A. Rodkina,
Almost sure asymptotic stability analysis of the $\theta$-Maruyama method applied to a test system with stabilising and destabilising stochastic perturbations, LMS J. Comp. Math., 15 (2012), 71-83.
doi: 10.1112/S1461157012000010. |
[2] |
E. Buckwar and T. Sickenberger,
A comparative linear mean-square stability analysis of Maruyama- and Milstein-type methods, Math. Comput. Simul., 81 (2011), 1110-1127.
doi: 10.1016/j.matcom.2010.09.015. |
[3] |
E. Buckwar and T. Sickenberger,
A structural analysis of asymptotic mean-square stability for multi-dimensional linear stochastic differential systems, Appl. Numer. Math., 62 (2012), 842-859.
doi: 10.1016/j.apnum.2012.03.002. |
[4] |
K. Burrage, I. Lenane and G. Lythe,
Numerical methods for second-order stochastic differential equations, SIAM J. Sci. Comput., 29 (2007), 245-264.
doi: 10.1137/050646032. |
[5] |
P. M. Burrage and K. Burrage,
Structure-preserving Runge-Kutta methods for stochastic Hamiltonian equations with additive noise, Numer. Algorithms, 65 (2014), 519-532.
doi: 10.1007/s11075-013-9796-6. |
[6] |
C. Chen, D. Cohen, R. D'Ambrosio and A. Lang, Drift-preserving numerical integrators for stochastic Hamiltonian systems, Adv. Comput. Math., 46 (2020), art. no. 27.
doi: 10.1007/s10444-020-09771-5. |
[7] |
V. Citro and R. D'Ambrosio,
Long-term analysis of stochastic $\theta$-methods for damped stochastic oscillators, Appl. Numer. Math., 150 (2020), 18-26.
doi: 10.1016/j.apnum.2019.08.011. |
[8] |
D. Conte, R. D'Ambrosio and B. Paternoster,
On the stability of $\vartheta$-methods for stochastic Volterra integral equations, Disc. Cont. Dyn. Sys. - Series B, 23 (2018), 2695-2708.
doi: 10.3934/dcdsb.2018087. |
[9] |
R. D'Ambrosio, G. Giordano, B. Paternoster and A. Ventola, Perturbative analysis of stochastic Hamiltonian problems under time discretizations, Appl. Math. Lett., 120 (2021), article number 107223.
doi: 10.1016/j.aml.2021.107223. |
[10] |
R. D'Ambrosio and S. D. Giovacchino,
Mean-square contractivivity of stochastic $\theta$-methods, Commun. Nonlinear Sci. Numer. Simul, 96 (2021), 105671.
doi: 10.1016/j.cnsns.2020.105671. |
[11] |
R. D'Ambrosio, M. Moccaldi and B. Paternoster,
Numerical preservation of long-term dynamics by stochastic two-step methods, Discr. Cont. Dyn. Sys. B, 23 (2018), 2763-2773.
doi: 10.3934/dcdsb.2018105. |
[12] |
R. D'Ambrosio and C. Scalone,
On the numerical structure preservation of nonlinear damped stochastic oscillators, Numer. Algorithm, 86 (2021), 933-952.
doi: 10.1007/s11075-020-00918-5. |
[13] |
H. de la Cruz, J. C. Jimenez and J. P. Zubelli,
Locally linearized methods for the sim- ulation of stochastic oscillators driven by random forces, BIT, 57 (2017), 123-151.
doi: 10.1007/s10543-016-0620-2. |
[14] |
G. Failla and A. Pirrotta,
On the stochastic response of a fractionally-damped Duffing oscillator, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 5131-5142.
doi: 10.1016/j.cnsns.2012.03.033. |
[15] |
C. W. Gardiner, Handbook of Stochastic Methods, for Physics, 3$^{rd}$ edition, Chemistry and the Natural Sciences, 13, Springer-Verlag, 2004. |
[16] |
M. B. Giles,
Multi-level Monte Carlo path simulation, Oper. Res., 56 (2008), 607-617.
doi: 10.1287/opre.1070.0496. |
[17] |
D. J. Higham,
Mean-square and asymptotic stability of the stochastic theta mehods, SIAM J. Numer. Anal., 38 (2000), 753-769.
doi: 10.1137/S003614299834736X. |
[18] |
D. J. Higham and P. Kloeden, An Introduction to the Numerical Simulation of Stochastic Differential Equations, SIAM, 2021. |
[19] |
P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992.
doi: 10.1007/978-3-662-12616-5. |
[20] |
K. Nouri, H. Ranjbar and J. C. Cortés López,
Modifying the split-step $\theta$-method with harmonic-mean term for stochastic differential equations, Int. J. Numer. Anal. Model., 17 (2020), 662-678.
|
[21] |
D. Roy,
A new numeric-analytical principle for nonlinear deterministic and stochastic dynamical systems, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 539-566.
doi: 10.1098/rspa.2000.0681. |
[22] |
Y. Saito and T. Mitsui,
Stability analysis of numerical schemes for stochastic differential equations, SIAM J. Numer. Anal., 33 (1996), 2254-2267.
doi: 10.1137/S0036142992228409. |
[23] |
J. Yalim, B. D. Welfert and J. M. Lopez,
Evaluation of closure strategies for a periodically-forced Duffing oscillator with slowly modulated frequency subject to Gaussian white noise, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 144-158.
doi: 10.1016/j.cnsns.2016.08.003. |
[24] |
J. Zhang and L. Wang,
A new symplectic method for a linear stochastic oscillator via stochastic variational integrators, AIP Conference Proceedings, 1479 (2012), 1772-1775.
doi: 10.1063/1.4756519. |
show all references
References:
[1] |
G. Berkolaiko, E. Buckwar, C. Kelly and A. Rodkina,
Almost sure asymptotic stability analysis of the $\theta$-Maruyama method applied to a test system with stabilising and destabilising stochastic perturbations, LMS J. Comp. Math., 15 (2012), 71-83.
doi: 10.1112/S1461157012000010. |
[2] |
E. Buckwar and T. Sickenberger,
A comparative linear mean-square stability analysis of Maruyama- and Milstein-type methods, Math. Comput. Simul., 81 (2011), 1110-1127.
doi: 10.1016/j.matcom.2010.09.015. |
[3] |
E. Buckwar and T. Sickenberger,
A structural analysis of asymptotic mean-square stability for multi-dimensional linear stochastic differential systems, Appl. Numer. Math., 62 (2012), 842-859.
doi: 10.1016/j.apnum.2012.03.002. |
[4] |
K. Burrage, I. Lenane and G. Lythe,
Numerical methods for second-order stochastic differential equations, SIAM J. Sci. Comput., 29 (2007), 245-264.
doi: 10.1137/050646032. |
[5] |
P. M. Burrage and K. Burrage,
Structure-preserving Runge-Kutta methods for stochastic Hamiltonian equations with additive noise, Numer. Algorithms, 65 (2014), 519-532.
doi: 10.1007/s11075-013-9796-6. |
[6] |
C. Chen, D. Cohen, R. D'Ambrosio and A. Lang, Drift-preserving numerical integrators for stochastic Hamiltonian systems, Adv. Comput. Math., 46 (2020), art. no. 27.
doi: 10.1007/s10444-020-09771-5. |
[7] |
V. Citro and R. D'Ambrosio,
Long-term analysis of stochastic $\theta$-methods for damped stochastic oscillators, Appl. Numer. Math., 150 (2020), 18-26.
doi: 10.1016/j.apnum.2019.08.011. |
[8] |
D. Conte, R. D'Ambrosio and B. Paternoster,
On the stability of $\vartheta$-methods for stochastic Volterra integral equations, Disc. Cont. Dyn. Sys. - Series B, 23 (2018), 2695-2708.
doi: 10.3934/dcdsb.2018087. |
[9] |
R. D'Ambrosio, G. Giordano, B. Paternoster and A. Ventola, Perturbative analysis of stochastic Hamiltonian problems under time discretizations, Appl. Math. Lett., 120 (2021), article number 107223.
doi: 10.1016/j.aml.2021.107223. |
[10] |
R. D'Ambrosio and S. D. Giovacchino,
Mean-square contractivivity of stochastic $\theta$-methods, Commun. Nonlinear Sci. Numer. Simul, 96 (2021), 105671.
doi: 10.1016/j.cnsns.2020.105671. |
[11] |
R. D'Ambrosio, M. Moccaldi and B. Paternoster,
Numerical preservation of long-term dynamics by stochastic two-step methods, Discr. Cont. Dyn. Sys. B, 23 (2018), 2763-2773.
doi: 10.3934/dcdsb.2018105. |
[12] |
R. D'Ambrosio and C. Scalone,
On the numerical structure preservation of nonlinear damped stochastic oscillators, Numer. Algorithm, 86 (2021), 933-952.
doi: 10.1007/s11075-020-00918-5. |
[13] |
H. de la Cruz, J. C. Jimenez and J. P. Zubelli,
Locally linearized methods for the sim- ulation of stochastic oscillators driven by random forces, BIT, 57 (2017), 123-151.
doi: 10.1007/s10543-016-0620-2. |
[14] |
G. Failla and A. Pirrotta,
On the stochastic response of a fractionally-damped Duffing oscillator, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 5131-5142.
doi: 10.1016/j.cnsns.2012.03.033. |
[15] |
C. W. Gardiner, Handbook of Stochastic Methods, for Physics, 3$^{rd}$ edition, Chemistry and the Natural Sciences, 13, Springer-Verlag, 2004. |
[16] |
M. B. Giles,
Multi-level Monte Carlo path simulation, Oper. Res., 56 (2008), 607-617.
doi: 10.1287/opre.1070.0496. |
[17] |
D. J. Higham,
Mean-square and asymptotic stability of the stochastic theta mehods, SIAM J. Numer. Anal., 38 (2000), 753-769.
doi: 10.1137/S003614299834736X. |
[18] |
D. J. Higham and P. Kloeden, An Introduction to the Numerical Simulation of Stochastic Differential Equations, SIAM, 2021. |
[19] |
P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992.
doi: 10.1007/978-3-662-12616-5. |
[20] |
K. Nouri, H. Ranjbar and J. C. Cortés López,
Modifying the split-step $\theta$-method with harmonic-mean term for stochastic differential equations, Int. J. Numer. Anal. Model., 17 (2020), 662-678.
|
[21] |
D. Roy,
A new numeric-analytical principle for nonlinear deterministic and stochastic dynamical systems, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 539-566.
doi: 10.1098/rspa.2000.0681. |
[22] |
Y. Saito and T. Mitsui,
Stability analysis of numerical schemes for stochastic differential equations, SIAM J. Numer. Anal., 33 (1996), 2254-2267.
doi: 10.1137/S0036142992228409. |
[23] |
J. Yalim, B. D. Welfert and J. M. Lopez,
Evaluation of closure strategies for a periodically-forced Duffing oscillator with slowly modulated frequency subject to Gaussian white noise, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 144-158.
doi: 10.1016/j.cnsns.2016.08.003. |
[24] |
J. Zhang and L. Wang,
A new symplectic method for a linear stochastic oscillator via stochastic variational integrators, AIP Conference Proceedings, 1479 (2012), 1772-1775.
doi: 10.1063/1.4756519. |


[1] |
Dajana Conte, Raffaele D'Ambrosio, Beatrice Paternoster. On the stability of $\vartheta$-methods for stochastic Volterra integral equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2695-2708. doi: 10.3934/dcdsb.2018087 |
[2] |
Seung-Yeal Ha, Dongnam Ko, Chanho Min, Xiongtao Zhang. Emergent collective behaviors of stochastic kuramoto oscillators. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1059-1081. doi: 10.3934/dcdsb.2019208 |
[3] |
Lijin Wang, Jialin Hong. Generating functions for stochastic symplectic methods. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 1211-1228. doi: 10.3934/dcds.2014.34.1211 |
[4] |
William F. Thompson, Rachel Kuske, Yue-Xian Li. Stochastic phase dynamics of noise driven synchronization of two conditional coherent oscillators. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2971-2995. doi: 10.3934/dcds.2012.32.2971 |
[5] |
Ying-Cheng Lai, Kwangho Park. Noise-sensitive measure for stochastic resonance in biological oscillators. Mathematical Biosciences & Engineering, 2006, 3 (4) : 583-602. doi: 10.3934/mbe.2006.3.583 |
[6] |
Jie Zhang, Yue Wu, Liwei Zhang. A class of smoothing SAA methods for a stochastic linear complementarity problem. Numerical Algebra, Control and Optimization, 2012, 2 (1) : 145-156. doi: 10.3934/naco.2012.2.145 |
[7] |
Martino Bardi, Annalisa Cesaroni, Daria Ghilli. Large deviations for some fast stochastic volatility models by viscosity methods. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 3965-3988. doi: 10.3934/dcds.2015.35.3965 |
[8] |
Lijin Wang, Pengjun Wang, Yanzhao Cao. Numerical methods preserving multiple Hamiltonians for stochastic Poisson systems. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 819-836. doi: 10.3934/dcdss.2021095 |
[9] |
Giulia Bertaglia, Liu Liu, Lorenzo Pareschi, Xueyu Zhu. Bi-fidelity stochastic collocation methods for epidemic transport models with uncertainties. Networks and Heterogeneous Media, 2022, 17 (3) : 401-425. doi: 10.3934/nhm.2022013 |
[10] |
Ying Liu, Yabing Sun, Weidong Zhao. Explicit multistep stochastic characteristic approximation methods for forward backward stochastic differential equations. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 773-795. doi: 10.3934/dcdss.2021044 |
[11] |
Raffaele D'Ambrosio, Martina Moccaldi, Beatrice Paternoster. Numerical preservation of long-term dynamics by stochastic two-step methods. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2763-2773. doi: 10.3934/dcdsb.2018105 |
[12] |
Christopher Rackauckas, Qing Nie. Adaptive methods for stochastic differential equations via natural embeddings and rejection sampling with memory. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2731-2761. doi: 10.3934/dcdsb.2017133 |
[13] |
Mei Ju Luo, Yi Zeng Chen. Smoothing and sample average approximation methods for solving stochastic generalized Nash equilibrium problems. Journal of Industrial and Management Optimization, 2016, 12 (1) : 1-15. doi: 10.3934/jimo.2016.12.1 |
[14] |
Ruilin Li, Xin Wang, Hongyuan Zha, Molei Tao. Improving sampling accuracy of stochastic gradient MCMC methods via non-uniform subsampling of gradients. Discrete and Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021157 |
[15] |
Xiaobing Feng, Shu Ma. Stable numerical methods for a stochastic nonlinear Schrödinger equation with linear multiplicative noise. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 687-711. doi: 10.3934/dcdss.2021071 |
[16] |
Tengteng Yu, Xin-Wei Liu, Yu-Hong Dai, Jie Sun. Variable metric proximal stochastic variance reduced gradient methods for nonconvex nonsmooth optimization. Journal of Industrial and Management Optimization, 2022, 18 (4) : 2611-2631. doi: 10.3934/jimo.2021084 |
[17] |
Quan Zhou, Yabing Sun. High order one-step methods for backward stochastic differential equations via Itô-Taylor expansion. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4387-4413. doi: 10.3934/dcdsb.2021233 |
[18] |
Yuri B. Gaididei, Carlos Gorria, Rainer Berkemer, Peter L. Christiansen, Atsushi Kawamoto, Mads P. Sørensen, Jens Starke. Stochastic control of traffic patterns. Networks and Heterogeneous Media, 2013, 8 (1) : 261-273. doi: 10.3934/nhm.2013.8.261 |
[19] |
John A. D. Appleby, Xuerong Mao, Alexandra Rodkina. On stochastic stabilization of difference equations. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 843-857. doi: 10.3934/dcds.2006.15.843 |
[20] |
Yu Wu, Xiaopeng Zhao, Mingjun Zhang. Dynamics of stochastic mutation to immunodominance. Mathematical Biosciences & Engineering, 2012, 9 (4) : 937-952. doi: 10.3934/mbe.2012.9.937 |
Impact Factor:
Tools
Article outline
Figures and Tables
[Back to Top]