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

作者：Przybylowicz Pawel^*

来源：Journal of Computational and Applied Mathematics, 2010, 235(1): 203-217.

DOI：10.1016/j.cam.2010.05.033

• Stochastic Ito integrals
• Singular problems
• Optimal algorithm
• Standard information
• r-fold integrated Brownian motion

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.

• 出版日期2010-11-1