Mean-square and asymptotic stability of the stochastic theta method
Mean-square and asymptotic stability of the stochastic theta method
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DOI：
10.1137/s003614299834736x
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发表时间：
2000-09-22
影响因子：
2.9
通讯作者：
Higham, DJ
中科院分区：
文献类型：
--
作者：
Higham, DJ
常微分方程数值方法的稳定性分析的动机是什么样的步长选择的问题，数值方法重现的测试方程的特性？本文研究了一个带乘性噪声项的线性检验方程，并考虑了随机形式的theta方法的均方稳定性和渐近稳定性。我们推广了[Saito and Mitsui，SIAM. J. Numer。分析：33（1996），pp. 2254-2267]。特别是，我们证明了确定性的A-稳定性属性的扩展。我们还绘制了均方稳定区域的情况下，测试方程有真实的参数。对于渐近稳定性，我们表明，这个问题减少到寻找一个参数化的随机变量的期望值。我们结合联合收割机的分析和数值技术，以获得洞察的稳定性。对于一个变量的方法，已提出在文献中，我们得到精确的解析表达式的渐近稳定区域。这使我们能够证明一些结果。所介绍的技术是广泛适用的，我们用它来表明，[Kloeden和Platen，随机微分方程的数值解，Springer-Verlag，1992]建议的全隐式方法具有确定性A-稳定性的渐近稳定性扩展。我们还使用该方法来解释一些数值结果中报道的[Milstein，Platen和Schurz，SIAM J. Numer.分析：35（1998），pp. 1010 1019.]。
Stability analysis of numerical methods for ordinary differential equations (ODEs) is motivated by the question for what choices of stepsize does the numerical method reproduce the characteristics of the test equation? We study a linear test equation with a multiplicative noise term, and consider mean-square and asymptotic stability of a stochastic version of the theta method. We extend some mean-square stability results in [Saito and Mitsui, SIAM. J. Numer. Anal., 33 (1996), pp. 2254-2267]. In particular, we show that an extension of the deterministic A-stability property holds. We also plot mean-square stability regions for the case where the test equation has real parameters. For asymptotic stability, we show that the issue reduces to finding the expected value of a parametrized random variable. We combine analytical and numerical techniques to get insights into the stability properties. For a varian of the method that has been proposed in the literature we obtain precise analytic expressions for the asymptotic stability region. This allows us to prove a number of results. The technique introduced is widely applicable, and we use it to show that a fully implicit method suggested by [Kloeden and Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, 1992] has an asymptotic stability extension of the deterministic A-stability property. We also use the approach to explain some numerical results reported in [Milstein, Platen, and Schurz, SIAM J. Numer. Anal., 35 (1998), pp. 1010 1019.].