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.
Highlights
Stochastic processes are used to model stochastic phenomena in various fields of science, engineering, economics and finance
Diffusion Processes (SDP), which have received considerable attention recently, due on the one hand to their diverse applications in stochastic modelling, and on the other, to their value in addressing probabilistic statistical problems, especially those involving statistical inference. These processes have been widely studied, and much research has been undertaken to resolve these issues of statistical inference, with particular respect to the estimation of parameters; see, among others, Bibby and Sorensen [1], Prakasa Rao [2], Chang and Cheng [3], Beskos et al [4], Stramer and Yan [5], Shoji and Ozaki [6], Durham and Gallant [7] and Fan [8], without forgetting the works of Yenkie and Diwekar [9] and Kloeden et al [10] and the important bibliography cited in these works
Let { X (t); t ∈ [t0, T ]; t0 ≥ 0} be a stochastic process taking values on (0, ∞), X (t) is a Gompertz diffusion process with parameters α, β and σ and which is denoted by Gomp(α; β; σ ) if X (t) satisfies
Summary
Stochastic processes are used to model stochastic phenomena in various fields of science, engineering, economics and finance. Gutiérrez et al [19,20,21], Ferrante et al [22], Román-Román et al [23] and Giorno and Nobile [24], have highlighted the importance of this process, and many subsequent extensions have appeared, especially regarding the non-homogeneous case with exogenous factors (external variables) that affect the drift coefficient These extensions take one of the following two forms: With external information (when no functional form is available): the exogenous factors are completely determined by the observed data (monthly, annual, etc.) and to obtain their functional forms interpolation methods, among others, can be used. The process and the methodology presented are applied to simulated data obtained from the explicit expression of the solution to the characteristic state equation for the process
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