# American Institute of Mathematical Sciences

doi: 10.3934/jcd.2021023
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## 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

* Corresponding author: Raffaele D'Ambrosio

Received  February 2021 Revised  August 2021 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.

Citation: Raffaele D'Ambrosio, Stefano Di Giovacchino. Numerical preservation issues in stochastic dynamical systems by $\vartheta$-methods. Journal of Computational Dynamics, 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] C. W. Gardiner, Handbook of Stochastic Methods, for Physics, 3$^{rd}$ edition, Chemistry and the Natural Sciences, 13, Springer-Verlag, 2004.  Google Scholar [16] M. B. Giles, Multi-level Monte Carlo path simulation, Oper. Res., 56 (2008), 607-617.  doi: 10.1287/opre.1070.0496.  Google Scholar [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.  Google Scholar [18] D. J. Higham and P. Kloeden, An Introduction to the Numerical Simulation of Stochastic Differential Equations, SIAM, 2021.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] C. W. Gardiner, Handbook of Stochastic Methods, for Physics, 3$^{rd}$ edition, Chemistry and the Natural Sciences, 13, Springer-Verlag, 2004.  Google Scholar [16] M. B. Giles, Multi-level Monte Carlo path simulation, Oper. Res., 56 (2008), 607-617.  doi: 10.1287/opre.1070.0496.  Google Scholar [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.  Google Scholar [18] D. J. Higham and P. Kloeden, An Introduction to the Numerical Simulation of Stochastic Differential Equations, SIAM, 2021.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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
Variance error growth with respect to the time $t$ computed by Euler-Maruyama method
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 $\omega_0 = 1.3$
 $\sigma$ $\Bigl|\mathbb{E}[\Theta_T] - \mathbb{E}[\Theta(T)]\Bigr|$ $\Bigl|\mathbb{V}[\Theta_T] - \mathbb{V}[\Theta(T)]\Bigr|$ $10^{-12}$ $3.98 \times 10^{-13}$ $5.26 \times 10^{-24}$ $10^{-9}$ $2.76 \times 10^{-10}$ $5.18 \times 10^{-18}$ $10^{-6}$ $2.76 \times 10^{-7}$ $5.18 \times 10^{-12}$ $10^{-3}$ $2.76 \times 10^{-4}$ $5.18 \times 10^{-6}$ $10^{0}$ $2.76 \times 10^{-1}$ $5.18 \times 10^{0}$
 $\sigma$ $\Bigl|\mathbb{E}[\Theta_T] - \mathbb{E}[\Theta(T)]\Bigr|$ $\Bigl|\mathbb{V}[\Theta_T] - \mathbb{V}[\Theta(T)]\Bigr|$ $10^{-12}$ $3.98 \times 10^{-13}$ $5.26 \times 10^{-24}$ $10^{-9}$ $2.76 \times 10^{-10}$ $5.18 \times 10^{-18}$ $10^{-6}$ $2.76 \times 10^{-7}$ $5.18 \times 10^{-12}$ $10^{-3}$ $2.76 \times 10^{-4}$ $5.18 \times 10^{-6}$ $10^{0}$ $2.76 \times 10^{-1}$ $5.18 \times 10^{0}$
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