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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.

Many scientific fields have experienced growth in the use of stochastic differential equations (SDEs), also known as diffusion processes, to model scientific phenomena over time. SDEs can simultaneously capture the known deterministic dynamics of underlying variables of interest (e.g., ocean flow, chemical and physical characteristics of a body of water, presence, absence, and spread of a disease), while enabling a modeler to capture the unknown random dynamics in a stochastic setting. We focus on reviewing a wide range of statistical inference methods for likelihood-based frequentist and Bayesian parametric inference based on discretely-sampled diffusions. Exact parametric inference is not usually possible because the transition density is not available in closed form. Thus, we review the literature on approximate numerical methods (e.g., Euler, Milstein, local linearization, and Aït-Sahalia) and simulation-based approaches (e.g., data augmentation and exact sampling) that are used to carry out parametric statistical inference on SDE processes. We close with a brief discussion of other methods of inference for SDEs and more complex SDE processes such as spatio-temporal SDEs.

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