Adaptive Ito-Taylor algorithm can optimally approximate the Ito integrals of singular functions

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:235

Article

Przybylowicz, Pawel

关键词: Stochastic Ito integrals;    Singular problems;    Optimal algorithm;    Standard information;    r-fold integrated Brownian motion;   

DOI  :  10.1016/j.cam.2010.05.033

来源: Elsevier

【 摘 要 】

We deal with numerical approximation of stochastic Ito integrals of singular functions. We first consider the regular case of integrands belonging to the Holder class with parameters r and e. We show that in this case the classical Ito-Taylor algorithm has the optimal error Theta(n(-(r+e))). In the singular case, we consider a class of piecewise regular functions that have continuous derivatives, except for a finite number of unknown singular points. We show that any nonadaptive algorithm cannot efficiently handle such a problem, even in the case of a single singularity. The error of such algorithm is no less than n(-min{1/2,r+e}). Therefore, we must turn to adaptive algorithms. We construct the adaptive Ito-Taylor algorithm that, in the case of at most one singularity, has the optimal error O(n(-(r+e))). The best speed of convergence, known for regular functions, is thus preserved. For multiple singularities, we show that any adaptive algorithm has the error Omega (n(-min{/2.r+e})), and this bound is sharp. (C) 2010 Elsevier B.V. All rights reserved.

【 授权许可】

Free