Statistical inference for stochastic differential equations

Abstract

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.