Parameter estimation and model selection for stochastic differential equations for biological growth

Abstract In this paper, we consider the stochastic versions of three classical growth models given by ordinary differential equations (ODEs). Indeed we use the stochastic versions of Gompertz, von Bertalanffy, and logistic differential equations as models. We assume that each stochastic differential equation (SDE) has some crucial parameters to be estimated, and we use maximum likelihood estimation (MLE) to estimate them. For estimating the diffusion parameter, we use the MLE for two cases and the quadratic variation of the data for one of the SDEs. We apply the Akaike information criterion (AIC) to choose the best model for the simulated data. We consider that the AIC is a function of the drift parameter. We conduct numerical experiments to validate our selection method. Subsequently, we also apply it to actual data. The proposed methodology could be applied to datasets with discrete observations, including highly sparse data. Indeed, we can use this method even in the extreme case where we have observed only one point for each path, under the condition that we observed a sufficient number of trajectories. For the last two cases, the data can be viewed as incomplete observations of a model with a tractable likelihood function; then, we propose a version of the expectation maximization (EM) algorithm to estimate these parameters. This type of dataset typically appears in fishery, for instance.