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The Invariance of Asymptotic Laws of Linear Stochastic Systems under Discretization
Author(s) -
Schurz H.
Publication year - 1999
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/(sici)1521-4001(199906)79:6<375::aid-zamm375>3.0.co;2-7
Subject(s) - mathematics , discretization , wiener process , white noise , continuous time stochastic process , stochastic process , multiplicative noise , parametric statistics , lyapunov function , stochastic differential equation , numerical integration , mathematical analysis , nonlinear system , computer science , statistics , physics , signal transfer function , digital signal processing , quantum mechanics , analog signal , computer hardware
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 corresponding 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.