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

• Authors: Paweł Przybyłowicz
• Venue: Journal of Computational and Applied Mathematics
• Year: 2010

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 @r. We show that in this case the classical Ito-Taylor algorithm has the optimal error @Q(n^-^(^r^+^@r^)). 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^-^m^i^n^{^1^/^2^,^r^+^@r^}. 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^+^@r^)). The best speed of convergence, known for regular functions, is thus preserved. For multiple singularities, we show that any adaptive algorithm has the error @W(n^-^m^i^n^{^1^/^2^,^r^+^@r^}), and this bound is sharp.