Simulating bridges using confluent diffusions

Abstract

Diffusions are a fundamental class of models in many fields, including finance, engineering, and biology. Simulating diffusions is challenging as their sample paths are infinite-dimensional and their transition functions are typically intractable. In statistical settings such as parameter inference for discretely observed diffusions, we require simulation techniques for diffusions conditioned on hitting a given endpoint, which introduces further complication. In this paper we introduce a Markov chain Monte Carlo algorithm for simulating bridges of ergodic diffusions which (i) is exact in the sense that there is no discretisation error, (ii) has computational cost that is linear in the duration of the bridges, and (iii) provides bounds on local maxima and minima of the simulated trajectory. Our approach works directly on diffusion path space, by constructing a proposal (which we term a confluence) that is then corrected with an accept/reject step in a pseudo-marginal algorithm. Our method requires only the simulation of unconditioned diffusion sample paths. We apply our approach to the simulation of Langevin diffusion bridges, a practical problem arising naturally in many situations, such as statistical inference in distributed settings.