Article Contents
Article Contents

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

• * Corresponding author: Raffaele D'Ambrosio

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.

Mathematics Subject Classification: 65C30.

 Citation:

• 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

Figure 2.  Variance error growth with respect to the time $t$ computed by Euler-Maruyama method

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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