Efficient almost-exact Lévy area sampling

Abstract

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 Chebyshev 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 1 0 − 12 . We thus establish a strong almost-exact Levy area sampling method. The complexity of our method is square logarithmic. We indicate how it can contribute to efficient sampling in higher dimensions.