In this paper, we present a method to adjust a stochastic logistic differential equation (SLDE) to a set of highly sparse real data. We assume that the SLDE have two unknown parameters to be estimated. We calculate the Maximum Likelihood Estimator (MLE) to estimate the intrinsic growth rate. We prove that the MLE is strongly consistent and asymptotically normal. For estimating the diffusion parameter, the quadratic variation of the data is used. We validate our method with several types of simulated data. For more realistic cases in which we observe discretizations of the solution, we use diffusion bridges and the stochastic expectation-maximization algorithm to estimate the parameters. Furthermore, when we observe only one point for each path for a given number of trajectories we were still able to estimate the parameters of the SLDE. As far as we know, this is the first attempt to fit stochastic differential equations (SDEs) to these types of data. Finally, we apply our method to real data coming from fishery. The proposed adjustment method can be applied to other examples of SDEs and is highly applicable in several areas of science, especially in situations of sparse data.
Keywords: EM algorithm; Stochastic logistic differential equation; biological growth; diffusion bridges.
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