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首页> 外文期刊>Zeitschrift fur Angewandte Mathematik und Mechanik >The Invariance of Asymptotic Laws of Linear Stochastic Systems under Discretization
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The Invariance of Asymptotic Laws of Linear Stochastic Systems under Discretization

机译：离散化线性随机系统渐近律的不变性

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摘要

The stochastic trapezoidal rule provides the only equidistant discretization scheme from the family of implicit Euler methods (see (12)) which possesses the same asymptotic (stationary) law as underlying continuous time, linear and autonomous stochastic systems with white or coloured noise. This identity holds even when integration time goes to infinity, independent of used integration step size! Especially, the asymptotic behaviour of first two moments of corre-sponding probability distributions is rigorously examined and compared in this paper. The coincidence of asymptotic moments is shown for autonomous systems with multiplicative (parametric) and additive noise using fixed point principles and the theory of positive operators. The key result turns out to be useful for adequate implementation of stochastic algorithms applied to numerical solution of autonomous stochastic differential equations. In particular, it has practical importance when accurate long time integration is required such as in the process of estimation of Lyapunov exponents or stationary measures for oscillators in mechanical engineering.
机译：随机梯形法则提供了隐式欧拉方法族（见（12））中唯一等距的离散化方案，该方案具有与潜在的连续时间，线性和自治的带有白色或彩色噪声的随机系统相同的渐近（平稳）定律。即使当积分时间达到无穷大时，该身份仍然成立，与所用积分步长无关！特别是，本文对相应概率分布的前两个时刻的渐近行为进行了严格的检验和比较。使用定点原理和正算子理论，显示了具有乘性（参数）和加性噪声的自治系统的渐近矩的重合。关键结果证明对于将随机算法适当地应用于自治随机微分方程的数值解是有用的。特别地，当需要精确的长时间积分时（例如在机械工程中的振荡器的Lyapunov指数估计或固定测量过程中），它具有实际意义。

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