arXiv:2611.07424v1 Announce Type: new Abstract: We study randomly shifted rank-1 lattice rules for computing $\mathbb{E}[g(\mathbf{X}_T)]$, where $\mathbf{X}_T$ is the terminal state of a stochastic differential equation (SDE) with constant diffusion matrix. Discretizing the SDE by the Euler--Maruyama scheme with $m$ uniform time steps turns the expectation into an integral whose dimension grows with $m$ and whose integrand has a nested composition structure; both complicate the lattice rule error analysis. We generate the Brownian path by the Brownian bridge construction and handle the nested structure using the multivariate Fa\`a di Bruno formula and a rooted-tree expansion. Under suitable regularity and growth conditions, we prove that an $N$-point randomly shifted lattice rule with a uniformly random generating vector achieves a mean absolute sampling error of $O(N^{-1+\varepsilon})$ for every $\varepsilon\in(0,1/2)$ and every prime $N$, with the implied constant independent of $m$. To improve stability and control higher error moments, the median lattice estimator is used. The estimator takes the median of $R=2k-1$ independent lattice estimates and achieves an $L^q$ error of $O(N^{-1+\varepsilon})$ for every $1\le qSource：https://arxiv.org/abs/2611.07424
My favorites