  This paper studies an approximation method for the log-likelihood function of
a nonlinear diffusion process using the bridge of the diffusion. The main
result (Theorem \refthm:approx) shows that this approximation converges
uniformly to the unknown likelihood function and can therefore be used
efficiently with any algorithm for sampling from the law of the bridge. We also
introduce an expected maximum likelihood (EML) algorithm for inferring the
parameters of discretely observed diffusion processes. The approach is
applicable to a subclass of nonlinear SDEs with constant volatility and drift
that is linear in the model parameters. In this setting, globally optimal
parameters are obtained in a single step by solving a linear system. Simulation
studies to test the EML algorithm show that it performs well when compared with
algorithms based on the exact maximum likelihood as well as closed-form
likelihood expansions.
