In this paper, we consider stochastic versions of three classical growth models given by ordinary differential equations (ODEs). Indeed we use stochastic versions of Von Bertalanffy, Gompertz, and Logistic differential equations as models. We assume that each stochastic differential equation (SDE) has some crucial parameters in the drift to be estimated and we use the Maximum Likelihood Estimator (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 present a simulation study to validate our selection method. The proposed methodology could be applied to datasets with continuous and discrete observations, but also with 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 datasets typically appears in fishery, for instance.

翻译：本文考虑由常微分方程给出的三种经典生长模型的随机版本。具体而言，我们采用冯·贝塔朗菲、冈珀茨和逻辑斯蒂微分方程的随机形式作为模型。假设每个随机微分方程在漂移项中均含有待估计的关键参数，并利用极大似然估计法对其进行估计。对于扩散参数，我们针对两种情况采用极大似然估计，而针对其中一种随机微分方程则采用数据的二次变差进行估计。我们运用赤池信息准则选择模拟数据的最优模型，并将AIC视为漂移参数的函数。通过模拟研究验证所提出的选择方法。该方法论可应用于连续观测、离散观测乃至高度稀疏的数据集。事实上，即使在每条路径仅观测到单个数据点的极端情形下，只要观测到足够数量的轨迹，该方法依然适用。对于后两种情况，数据可视为具有可处理似然函数模型的不完全观测；为此，我们提出一种基于期望最大化算法的参数估计方案。这类数据集通常出现在渔业等研究领域中。