Higher-order stochastic integration through cubic stratification

We propose two novel unbiased estimators of the integral ${\int }_{[0,1{]}^s}f(u)du$ for a function $f$, which depend on a smoothness parameter $r\in \mathbb{N}$. The first estimator integrates exactly the polynomials of degrees $p<r$ and achieves the optimal error $n^{-1\mathrm{/}2-r\mathrm{/}s}$ (where $n$ is the number of evaluations of $f$) when $f$ is $r$ times continuously differentiable. The second estimator is computationally cheaper but it is restricted to functions that vanish on the boundary of $[0,1{]}^s$. The construction of the two estimators relies on a combination of cubic stratification and control ariates based on numerical derivatives. We provide numerical evidence that they show good performance even for moderate values of $n$.