  With a view to statistical inference for discretely observed diffusion
models, we propose simple methods of simulating diffusion bridges,
approximately and exactly. Diffusion bridge simulation plays a fundamental role
in likelihood and Bayesian inference for diffusion processes. First a simple
method of simulating approximate diffusion bridges is proposed and studied.
Then these approximate bridges are used as proposal for an easily implemented
Metropolis-Hastings algorithm that produces exact diffusion bridges. The new
method utilizes time-reversibility properties of one-dimensional diffusions and
is applicable to all one-dimensional diffusion processes with finite
speed-measure. One advantage of the new approach is that simple simulation
methods like the Milstein scheme can be applied to bridge simulation. Another
advantage over previous bridge simulation methods is that the proposed method
works well for diffusion bridges in long intervals because the computational
complexity of the method is linear in the length of the interval. For
$\rho$-mixing diffusions the approximate method is shown to be particularly
accurate for long time intervals. In a simulation study, we investigate the
accuracy and efficiency of the approximate method and compare it to exact
simulation methods. In the study, our method provides a very good approximation
to the distribution of a diffusion bridge for bridges that are likely to occur
in applications to statistical inference. To illustrate the usefulness of the
new method, we present an EM-algorithm for a discretely observed diffusion
process.
