Efficient almost-exact Levy area sampling

TL;DR

This paper introduces a new, efficient method for sampling the Levy area in two-dimensional Wiener processes, enabling more accurate stochastic differential equation simulations with controlled error and low complexity.

Contribution

It presents a novel almost-exact Levy area sampling technique using infinite sums of Logistic variables and Chebychev polynomial approximation, improving efficiency and accuracy.

Findings

Achieves uniform error of 10^(-12) in Levy area sampling.

Complexity is logarithmic squared, making it computationally efficient.

Method is adaptable for higher-dimensional sampling.

Abstract

We present a new method for sampling the Levy area for a two-dimensional Wiener process conditioned on its endpoints. An efficient sampler for the Levy area is required to implement a strong Milstein numerical scheme to approximate the solution of a stochastic differential equation driven by a two-dimensional Wiener process whose diffusion vector fields do not commute. Our method is simple and complementary to those of Gaines-Lyons and Wiktorsson, and amenable to quasi-Monte--Carlo implementation. It is based on representing the Levy area by an infinite weighted sum of independent Logistic random variables. We use Chebychev polynomials to approximate the inverse distribution function of sums of independent Logistic random variables in three characteristic regimes. The error is controlled by the degree of the polynomials, we set the error to be uniformly 10^(-12). We thus establish a strong almost-exact Levy area sampling method. The complexity of our method is square logarithmic. We indicate how our method can contribute to efficient sampling in higher dimensions.