Many fields have experienced a recent growth in the use of stochastic differential equations (SDEs) to model scientific phenomena over time. Examples include applications in oceanography, ecology, and public health. SDEs can simultaneously capture the known deterministic dynamics of the variables of interest (e.g., ocean flow, the chemical and physical characteristics of a body of water, the presence, absence and spread of a disease), while enabling a modeler to capture the unknown dynamics and measurement processes in a stochastic setting. This proposal develops statistical methodology for building, fitting, and diagnosing the fit of multivariate and spatially-varying SDEs. Such models, which are often inspired by mechanistic modeling, can incorporate the complex dynamics of the variables of interest.
Although statistical methods for the fitting and analysis of SDEs models using data at a single location are becoming more widely used, accurate statistical methods for multivariate SDEs and SDEs indexed in space are far less developed. This project will derive improved approximate methods of inference for one- and multi-dimensional SDEs that are more accurate than the commonly used, but naive, Euler approximations. Spatially-varying SDE models will be built for modeling spatio-temporal data observed potentially irregularly in space and time. This research will be applied to address problems in applied disciplines. Education of students in statistical methods for SDEs will be an important goal of this project. In addition, a number of outreach programs will be used to educate a broader audience.
