*Article* **Powers of the Stochastic Gompertz and Lognormal Diffusion Processes, Statistical Inference and Simulation**

#### **Eva María Ramos-Ábalos 1,\*,†, Ramón Gutiérrez-Sánchez 1,† and Ahmed Nafidi 2,†**


Received: 5 March 2020; Accepted: 13 April 2020; Published: 15 April 2020

**Abstract:** 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.

**Keywords:** powers of stochastic Gompertz diffusion models; powers of stochastic lognormal diffusion models; estimation in diffusion process; stationary distribution and ergodicity; trend function; application to simulated data
