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

We deal with numerical approximation of stochastic Itô integrals of singular functions. We first consider the regular case of integrands belonging to the Hölder class with parameters rr and ϱϱ. We show that in this case the classical Itô–Taylor algorithm has the optimal error Θ(n−(r+ϱ))Θ(n−(r+ϱ)). 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+ϱ}n−min{1/2,r+ϱ}. Therefore, we must turn to adaptive algorithms. We construct the adaptive Itô–Taylor algorithm that, in the case of at most one singularity, has the optimal error O(n−(r+ϱ))O(n−(r+ϱ)). The best speed of convergence, known for regular functions, is thus preserved. For multiple singularities, we show that any adaptive algorithm has the error Ω(n−min{1/2,r+ϱ})Ω(n−min{1/2,r+ϱ}), and this bound is sharp.