ARCANE: Scalable high-degree cubature formulae for simulating SDEs without Monte Carlo error

TL;DR

ARCANE introduces an efficient algorithm to construct high-degree cubature formulae for SDE simulation, significantly reducing error compared to Monte Carlo methods without requiring large sample sizes.

Contribution

The paper presents ARCANE, a novel algorithm capable of automatically generating high-degree cubature formulae for SDEs, surpassing previous degree limitations and improving simulation accuracy.

Findings

Reaches degree D=19 cubature formulae within hours.

Achieves error reduction orders of magnitude over Monte Carlo.

Efficiently constructs cubature formulae on modest hardware.

Abstract

Monte Carlo sampling is the standard approach for estimating properties of solutions to stochastic differential equations (SDEs), but accurate estimates require huge sample sizes. Lyons and Victoir (2004) proposed replacing independently sampled Brownian driving paths with "cubature formulae", deterministic weighted sets of paths that match Brownian "signature moments" up to some degree $D$. They prove that cubature formulae exist for arbitrary $D$, but explicit constructions are difficult and have only reached $D=7$, too small for practical use. We present ARCANE, an algorithm that efficiently and automatically constructs cubature formulae of arbitrary degree. It reproduces the state of the art in seconds and reaches $\boldsymbol{D=19}$ within hours on modest hardware. In simulations across multiple different SDEs and error metrics, our cubature formulae robustly achieve an error orders of magnitude smaller than Monte Carlo with the same number of paths.