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Tractability of multi-parametric Euler and Wiener integrated processes

Lifshits M. A., Papageorgiou A., Woźniakowski H.

Probability and Mathematical Statistics

2012

Vol. 32, Fasc. 1

131--165

We study average case approximation of Euler and Wiener integrated processes of d variables which are almost surely rk-times continuously differentiable with respect to the k-th variable and 0 ≤ rk ≤ rk+1. Let n(ε; d) denote the minimal number of continuous linear functionals which is needed to find an algorithm that uses n such functionals and whose average case error improves the average case error of the zero algorithm by a factor ε. Strong polynomial tractability means that there are nonnegative numbers C and p such that [formula]. We prove that the Wiener process is much more difficult to approximate than the Euler process. Namely, strong polynomial tractability holds for the Euler case iff ...[formula]. Other types of tractability are also studied.