With a view to statistical inference for discretely observed diffusionmodels, we propose simple methods of simulating diffusion bridges,approximately and exactly. Diffusion bridge simulation plays a fundamental rolein likelihood and Bayesian inference for diffusion processes. First a simplemethod of simulating approximate diffusion bridges is proposed and studied.Then these approximate bridges are used as proposal for an easily implementedMetropolis-Hastings algorithm that produces exact diffusion bridges. The newmethod utilizes time-reversibility properties of one-dimensional diffusions andis applicable to all one-dimensional diffusion processes with finitespeed-measure. One advantage of the new approach is that simple simulationmethods like the Milstein scheme can be applied to bridge simulation. Anotheradvantage over previous bridge simulation methods is that the proposed methodworks well for diffusion bridges in long intervals because the computationalcomplexity of the method is linear in the length of the interval. For$\rho$-mixing diffusions the approximate method is shown to be particularlyaccurate for long time intervals. In a simulation study, we investigate theaccuracy and efficiency of the approximate method and compare it to exactsimulation methods. In the study, our method provides a very good approximationto the distribution of a diffusion bridge for bridges that are likely to occurin applications to statistical inference. To illustrate the usefulness of thenew method, we present an EM-algorithm for a discretely observed diffusionprocess.
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