Maruyama型和Milstein型方法的线性均方稳定性比较分析
A comparative linear mean-square stability analysis of Maruyama- and Milstein-type methods
AI总结：
本文通过扩展的线性测试方程，比较了Theta-Maruyama和Theta-Milstein方法在求解随机微分方程时的均方稳定性，发现Milstein型方法的稳定性条件更严格且依赖于噪声项，并引入部分隐式扩散近似以控制稳定性。
摘要：
本文比较了用于求解随机微分方程的Theta-Maruyama和Theta-Milstein方法的均方稳定性。在线性稳定性分析中，我们提出将标准几何布朗运动扩展为测试方程，并考虑一个带有多个乘性噪声项的标量线性测试方程。该测试方程允许在分析仍可处理的情况下，开始研究多维噪声对方法稳定性行为的影响。我们的发现包括：(i) Theta-Milstein方法的稳定性条件，因此对于某些Theta的选择，步长条件比Theta-Maruyama方法严格得多；(ii) Theta-Milstein方法的精确稳定区域显式依赖于噪声项。此外，我们研究了在Milstein型方法的扩散近似项中引入部分隐式性的效果，从而通过另一个方法参数Sigma获得控制这些方法稳定性的可能性。数值例子说明了结果，并比较了不同方法的稳定性行为。
英文摘要：
In this article we compare the mean-square stability properties of the Theta-Maruyama and Theta-Milstein method that are used to solve stochastic differential equations. For the linear stability analysis, we propose an extension of the standard geometric Brownian motion as a test equation and consider a scalar linear test equation with several multiplicative noise terms. This test equation allows to begin investigating the influence of multi-dimensional noise on the stability behaviour of the methods while the analysis is still tractable. Our findings include: (i) the stability condition for the Theta-Milstein method and thus, for some choices of Theta, the conditions on the step-size, are much more restrictive than those for the Theta-Maruyama method; (ii) the precise stability region of the Theta-Milstein method explicitly depends on the noise terms. Further, we investigate the effect of introducing partially implicitness in the diffusion approximation terms of Milstein-type methods, thus obtaining the possibility to control the stability properties of these methods with a further method parameter Sigma. Numerical examples illustrate the results and provide a comparison of the stability behaviour of the different methods.
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