  Stochastic Differential Equations (SDEs) are used as statistical models in
many disciplines. However, intractable likelihood functions for SDEs make
inference challenging, and we need to resort to simulation-based techniques to
estimate and maximize the likelihood function. While sequential Monte Carlo
methods have allowed for the accurate evaluation of likelihoods at fixed
parameter values, there is still a question of how to find the maximum
likelihood estimate. In this article we propose an efficient
Gaussian-process-based method for exploring the parameter space using estimates
of the likelihood from a sequential Monte Carlo sampler. Our method accounts
for the inherent Monte Carlo variability of the estimated likelihood, and does
not require knowledge of gradients. The procedure adds potential parameter
values by maximizing the so-called expected improvement, leveraging the fact
that the likelihood function is assumed to be smooth. Our simulations
demonstrate that our method has significant computational and efficiency gains
over existing grid- and gradient-based techniques. Our method is applied to
modeling the closing stock price of three technology firms.
