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

Raffaele D'Ambrosio; Stefano Di Giovacchino

doi:10.3934/jcd.2021023

Article Contents

2022, Volume 9, Issue 2: 123-131. Doi: 10.3934/jcd.2021023

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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
^* Corresponding author: Raffaele D'Ambrosio

Received: January 31, 2021

Revised: July 31, 2021

Published: April 2022

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

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Abstract

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:

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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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