  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.
