Continuous-Time Modeling of Random Searches: Statistical Properties and Inference

Mathematical modeling of random searches is of great relevance in the fields of physics, chemistry, biology or modern ecology. A large number of existing studies record the search movement at equidistant time intervals and model such time series data directly with discrete-time random walks, such as Lévy flights and correlated random walks. Given the increasing availability of high-resolution observation data, statistical inference for search paths based on such high-resolution data has recently become one of the major interests and has raised an important issue of robustness of random walk models to the sampling rate, as estimation results for the discrete observation data are found to be largely different at different sampling rates even when the underlying movement is supposedly independent of scale. To address this issue, in this paper, we propose to model the continuous-time search paths directly with a continuous-time stochastic process for which the observer makes statistical inference based on its discrete observation. As continuous-time counterparts of Lévy flights, we consider two-dimensional Lévy processes and discuss the relevance of those models based upon advantages and limitations in terms of statistical properties and inference. Among the proposed models, the Brownian motion is most tractable in various ways while its Gaussianity and infinite variation of sample paths do not well describe the reality. Such drawbacks in statistical properties may be remedied by employing the stable and tempered stable Lévy motions, while those models are less tractable and cause an issue in statistical inference.