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April  2022, 9(2): 123-131. doi: 10.3934/jcd.2021023

Numerical preservation issues in stochastic dynamical systems by $ \vartheta $-methods

Raffaele D'Ambrosio , and  Stefano Di Giovacchino

Department of Information Engineering and Computer Science and Mathematics, University of L'Aquila, Via Vetoio, Loc. Coppito - 67100 L'Aquila, Italy

* Corresponding author: Raffaele D'Ambrosio

Received  February 2021 Revised  August 2021 Published  April 2022 Early access  December 2021

Fund Project: This work is supported by GNCS-INDAM project and by PRIN2017-MIUR project 2017JYCLSF entitled "Structure preserving approximation of evolutionary problems". The authors are member of the INDAM Research group GNCS

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.

Keywords: Stochastic oscillators, stochastic $ \vartheta $-methods.
Mathematics Subject Classification: 65C30.
Citation: Raffaele D'Ambrosio, Stefano Di Giovacchino. Numerical preservation issues in stochastic dynamical systems by $ \vartheta $-methods. Journal of Computational Dynamics, 2022, 9 (2) : 123-131. doi: 10.3934/jcd.2021023
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.

Figure 1.  Variance error growth with respect to $ \sigma $ computed by Euler-Maruyama method, that is (8) with $ \vartheta = 0 $, at $ T = 100 $, computed with respect to $ 1000 $ paths
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Figure 2.  Variance error growth with respect to the time $ t $ computed by Euler-Maruyama method
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Table 1.  Growth of the errors for the mean and the variance of $ \Theta(T) $ computed by the stochastic trapezoidal rule with $ T = 100 $, $ M = 1000 $, $ h = 10^{-2} $, $ \varepsilon = 10^{-3} $ and $ \omega_0 = 1.3 $
$ \sigma $ $ \Bigl|\mathbb{E}[\Theta_T] - \mathbb{E}[\Theta(T)]\Bigr| $ $ \Bigl|\mathbb{V}[\Theta_T] - \mathbb{V}[\Theta(T)]\Bigr| $
$ 10^{-12} $ $ 3.98 \times 10^{-13} $ $ 5.26 \times 10^{-24} $
$ 10^{-9} $ $ 2.76 \times 10^{-10} $ $ 5.18 \times 10^{-18} $
$ 10^{-6} $ $ 2.76 \times 10^{-7} $ $ 5.18 \times 10^{-12} $
$ 10^{-3} $ $ 2.76 \times 10^{-4} $ $ 5.18 \times 10^{-6} $
$ 10^{0} $ $ 2.76 \times 10^{-1} $ $ 5.18 \times 10^{0} $
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$ \sigma $ $ \Bigl|\mathbb{E}[\Theta_T] - \mathbb{E}[\Theta(T)]\Bigr| $ $ \Bigl|\mathbb{V}[\Theta_T] - \mathbb{V}[\Theta(T)]\Bigr| $
$ 10^{-12} $ $ 3.98 \times 10^{-13} $ $ 5.26 \times 10^{-24} $
$ 10^{-9} $ $ 2.76 \times 10^{-10} $ $ 5.18 \times 10^{-18} $
$ 10^{-6} $ $ 2.76 \times 10^{-7} $ $ 5.18 \times 10^{-12} $
$ 10^{-3} $ $ 2.76 \times 10^{-4} $ $ 5.18 \times 10^{-6} $
$ 10^{0} $ $ 2.76 \times 10^{-1} $ $ 5.18 \times 10^{0} $
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