Angaben aus der Verlagsmeldung

Random Ordinary Differential Equations and their Numerical Solution / von Xiaoying Han, Peter E. Kloeden

<p>This book makes the recent results on the derivation of higher order numerical schemes for random ordinary differential equations (RODEs) available to a wider audience, and raises awareness of RODEs themselves and the closely associated theory of random dynamical systems, the latter through well-chosen illustrative examples. It also shows how RODEs are being used in the biological sciences, where non-Gaussian and bounded noise are often more realistic than the Gaussian white noise in SODEs.</p>Random ordinary differential equations (RODEs) are ordinary differential equations (ODEs) that include a stochastic process in their vector field. They are used in numerous important applications and play a fundamental role in the theory of random dynamical systems. Unlike stochastic differential equations (SODEs), RODEs can be analysed pathwise with deterministic calculus, but require further treatment beyond that of classical ODE theory due to the lack of smoothness in their temporal variable. Thus the solutions of RODEs do not have sufficient smoothness to have Taylor expansions in the usual sense.<p></p><p> Although classical numerical schemes for ODEs can be used pathwise for RODEs, they rarely attain their traditional order. As for Itô SODEs, Taylor expansions can be derived using an iterated application of the appropriate chain rule in integral form and are the starting point for the systematic derivation of consistent higher order numerical schemes for RODEs.<br /></p>