Powers of the Stochastic Gompertz and Lognormal Diffusion Processes, Statistical Inference and Simulation
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Powers of stochastic Gompertz diffusion modelsPowers of stochastic lognormal diffusion modelsEstimation in diffusion processStationary distribution and ergodicityTrend functionApplication to simulated data
Ramos-Ábalos, E. M., Gutiérrez-Sánchez, R., & Nafidi, A. (2020). Powers of the Stochastic Gompertz and Lognormal Diffusion Processes, Statistical Inference and Simulation. Mathematics, 8(4), 588. [doi:10.3390/math8040588]
SponsorshipThis research has been funded by “Programa Operativo FEDER de Andalucía 2014-2020 A-FQM228-UGR18”
In this paper, we study a new family of Gompertz processes, defined by the power of the homogeneous Gompertz diffusion process, which we term the powers of the stochastic Gompertz diffusion process. First, we show that this homogenous Gompertz diffusion process is stable, by power transformation, and determine the probabilistic characteristics of the process, i.e., its analytic expression, the transition probability density function and the trend functions. We then study the statistical inference in this process. The parameters present in the model are studied by using the maximum likelihood estimation method, based on discrete sampling, thus obtaining the expression of the likelihood estimators and their ergodic properties. We then obtain the power process of the stochastic lognormal diffusion as the limit of the Gompertz process being studied and go on to obtain all the probabilistic characteristics and the statistical inference. Finally, the proposed model is applied to simulated data.