Exact Simulation for Multivariate Itô Diffusions
Jose Blanchet, Fan Zhang
TL;DR
This work addresses the challenge of exact simulation for multidimensional Itô diffusions beyond the Lamperti-transformation regime. It develops two complementary strategies: (i) exact sampling for SDEs with identity diffusion using Tolerance-Enforced Simulation and localization of the Girsanov likelihood, and (ii) a general diffusion framework that leverages a Multilevel representation of the transition density, coupled with ancillary variables and Bernoulli factories to enable unbiased sampling via acceptance-rejection. The key contributions are a generic exact-sampling algorithm for multidimensional diffusions and a novel integration of rough-path concepts with Multilevel Monte Carlo to construct unbiased density estimators and exact samples. While the approach is primarily theoretical due to infinite expected termination time, it establishes a foundational framework for unbiased sampling in high-dimensional SDEs and suggests paths toward finite-time or controlled-bias variants with practical benefits in parallel computing and stochastic optimization.
Abstract
We provide the first generic exact simulation algorithm for multivariate diffusions. Current exact sampling algorithms for diffusions require the existence of a transformation which can be used to reduce the sampling problem to the case of a constant diffusion matrix and a drift which is the gradient of some function. Such transformation, called Lamperti transformation, can be applied in general only in one dimension. So, completely different ideas are required for exact sampling of generic multivariate diffusions. The development of these ideas is the main contribution of this paper. Our strategy combines techniques borrowed from the theory of rough paths, on one hand, and multilevel Monte Carlo on the other.