Exact Simulation of Multivariate Diffusions and Bridges

2026-10-08

AI Summary

This study addresses the challenge that existing exact sampling algorithms for multivariate diffusion processes and bridges lack finite expected total cost. To overcome this, we propose the first general-purpose exact sampling framework tailored to uniformly elliptic multivariate diffusions and their bridges. Methodologically, we construct a computational cost model based on random variable generation and function evaluations, integrating randomized parametrix expansions, acceptance-rejection sampling, and endpoint-weighted graph structures. Theoretically, this work achieves the first general exact sampling with finite expected cost. Furthermore, we establish minimax optimality for diffusion sampling over the joint dimension of time horizon and number of observations, as well as minimax optimality for bridge sampling under fixed conditions.

Abstract

We provide the first generic exact sampling algorithms with finite expected total cost for uniformly elliptic multivariate diffusions and the corresponding diffusion bridges with bounded coefficients and bounded continuous first derivatives. We formulate a computational cost model that counts work in terms of generation of Gaussian and uniform random variables to execute acceptance--rejection sampling, as well as function evaluations of the drift and diffusion coefficients, together with elementary operations such as sums and multiplications. Our constructions combine randomized parametrix expansions with a finite weighted graph that incorporates both bridge endpoints into the proposal. The exact sampling of the diffusion is minimax optimal jointly in the horizon and number of requested observations, while bridge sampling is minimax optimal in the number of observations for fixed horizon and endpoints.

Problem

• Multivariate Diffusions
• Diffusion Bridges
• Exact Simulation
• Sampling Algorithms

Innovation

• Exact Simulation
• Multivariate Diffusions
• Diffusion Bridges
• Randomized Parametrix Expansions
• Minimax Optimality