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

: 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 simulated


Introduction
Stochastic processes are used to model stochastic phenomena in various fields of science, engineering, economics and finance. An important category among these processes is that of Stochastic 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. In consequence, 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.
There has been much recent interest in applying SDP, and many researchers are working on the construction of stochastic processes in order to model phenomena of interest. These processes are used in areas such as the stochastic economy, new technologies, interest rates, courses of action, insurance, finance in general, cell growth, radiotherapy, chemotherapy, emissions from energy consumption and the emissions of CO 2 and greenhouse gases. Research results have been applied to various processes,

An Overview of the Homogeneous Gompertz Stochastic Diffusion Process
Let {X(t); t ∈ [t 0 , T]; t 0 ≥ 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 Ito's Stochastic Differential Equation (SDE) as follows (see [16,18,20,37]): In the literature, the constant α (∈ R) is the intrinsic growth rate; the β (∈ R) constant is the deceleration factor, the σ > 0 constant is the diffusion coefficient, X t 0 > 0 is a fixed real number and w t denotes the one-dimensional standard Wiener process.
The analytical expression of the unique solution to Equation (1) is given by (see, for example, [21,37]) From this, we deduce that the process X(t) is distributed as the following one-dimensional lognormal distribution: It has been shown (see [21]), that for β > 0, X(t) is ergodic and that the stationary distribution has a lognormal distribution. Hence, we have:
By applying Ito's formula to the transform given in Equation (4), we have Then, after some algebraic rearrangement, we obtain This shows that the process x γ (t) is also a Gomp(a; β; c) process, where: and c = γσ and the drift and diffusion coefficients are given respectively by: The model proposed in this paper belongs to the family of processes γ-PSGDP {x γ (t); t ∈ [t 0 , T]; t 0 ≥ 0} defined by the following SDE:

Probabilistic Characteristics of the γ-PSGDP
Under the initial condition given, the unique solution of the SDE Equation (5) can be obtained using the relations expressed by Equations (2) and (4), from which we have We then deduce that x γ (t) is distributed as a one dimensional lognormal distribution Λ 1 (µ(s, t, x t 0 ), γ 2 σ 2 λ 2 (t 0 , t)), where µ(s, t, x t 0 ) and λ 2 (t 0 , t) are given by From the homogeneity of the process, we know that x γ (t) | x γ (s) = x s has the lognormal distribution Λ 1 (µ(s, t, x s ), σ 2 λ 2 (s, t)), and then the PTDF of the process is The rth conditional moment of the process is given by from which the Conditional Trend Function (CTF) gives Assuming the initial condition P(x γ (t 0 ) = x t 0 ) = 1, the Trend Function (TF) of the process is From Equation (3), we deduce that for β > 0, the stationary distribution of the process is also a lognormal distribution and thus we have: Therefore, the asymptotic trend function of the process (for β > 0) is given by The limit of the trend function in Equation (7) (when t tends to ∞) coincides with this asymptotic trend function.

Likelihood Parameter Estimation
In the present study, with discrete sampling, we estimate the parameters α, σ 2 and β of the model by applying Maximum Likelihood (ML) methodology, following the same scheme as in Gutiérrez et al. [21]. To do so, we consider a discrete sampling of the process x γ (t 1 ) = x 1 , x γ (t 2 ) = x 2 , . . . , x γ (t n ) = x n for times t 1 , t 2 , . . . , t n and assume, moreover, that the length of the time intervals Then the associated likelihood function can be obtained by the following expression: The variable change can be used to work with a known probability function and to calculate the maximum probability estimators in a simpler way, considering the following transformation: . . , n and denoting V β = (v 2,β , . . . , v n,β ) . Thus, in terms of V β , the likelihood function is expressed as follows: (1 − e −2hβ ) and U = (1, . . . , 1) is a vector of the order (n − 1).
By differentiating the log-likelihood function with respect to a γ and c 2 γ , we obtain the following equations: The third likelihood equation is obtained by differentiating the log-likelihood function with respect to β and by using the effect that . . , log(x n )) and I x = (log(x 1 ), . . . , log(x n−1 )) . After various operations, we have Taking into account that U U = n − 1 and after algebraic rearrangement (not shown), the ML estimators of a γ and c 2 γ are The ML estimator of β is given byβ where H U = I n−1 − 1 n−1 UU is idempotent and a symmetric matrix and I n−1 denotes the identity matrix.

Asymptotic Properties of the Parameter Drift Estimators
Let X be a random variable with a distribution function given by Equation (8); then log(X) is distributed as a normal distribution N 1 . If β > 0, the process under consideration has ergodic properties, and for θ * = (a γ , β) ∈ (a γ,1 , a γ,2 ) × (β 1 , β 2 ), with β 1 > 0, we have I(θ) is the information matrix and is given by Then, we have and the inverse is An approximated, asymptotic confidence region of θ and an approximated, asymptotic marginal confidence interval of α and β can be obtained from Equations (12) and (13). The above-mentioned region is given, for a large T, by P T θ −θ * Î (θ) θ −θ ≤ χ 2 2,ξ = 1 − ξ obtaining I(θ) by replacing the parameters by their estimators and where χ 2 2,ξ represents the upper 100ξ per cent points of the chi squared distribution with two degrees of freedom.
The ξ% confidence (marginal) intervals for parameters α and β are given, for a large T, by where λ ξ represents the 100ξ per cent points of the normal standard distribution.
Note that in Equations (14) and (15) we have assumed that σ is known with a value σ =σ.

Powers of the Lognormal Diffusion Process
The Stochastic Lognormal Diffusion Process (SLDP) is known to be a particular case of the Gompertz diffusion process when the deceleration factor β = 0 (see, for example [21]). Then, the power of the SLDP can be obtained from that of the SGDP by tending β to zero.
Then, if the SLDP Y(t) is given by the following SDE: The resulting γ-PSLDP (y γ (t) = Y γ (t)) is governed by the following SDE: The same approach can be used to derive all the probabilistic properties and statistics for the γ-PSLDP process, taking β = 0 on the perspective equations established for the properties of γ-PSGDP in the previous sections, except as regards the symptotic properties of the drift parameter estimators (we already know that there is no asymptotic distribution in the case of the SLDP). For the latter case, we can obtain the exact distributions of the estimators, together with the confidence intervals for the process parameters (see [21]).

Estimated Trend Functions
In the same way as in Gutiérrez et al. [21], by Zehna's theorem [38], the Estimated Conditional Trend (ECT) and the Estimated Trend (ET) functions can be obtained from Equations (6) and (7) by replacing the parameters by their estimators. Furthermore, we can obtain an approximated and asymptotic confidence interval of the ETF and ECTF by means of the approximated and asymptotic confidence interval of the parameters given by Equations (14) and (15).

Simulation and Application
The trajectory of the model can be obtained by simulating the exact solution of SDE Equation (4) obtained in Equation (5). From this explicit solution, the simulated trajectories of the process are obtained from the following discretising time interval [t 0 , T]: t i = t 0 + ih, for i = 1, . . . , N (N is an integer and h is the discretization step), taking into account that the random variable in the latter expression σ(w t ) − w(t 1 ) is distributed as a one-dimensional normal distribution N (0, σ 2 (t − t 1 )) ( [39]). Table 1 shows the simulated data and the ETF for different powers, considering h = 1, N = 30, and the initial value x 1 = 0.99. We estimate the parameters by maximum likelihood, reserving the values observed for the time t = 30 for comparison with the corresponding prediction by the model. The results are shown in Table 2.  Figure 1 shows the fit and the prediction obtained for x γ (t) using the ETF (γ = 1 γ = 1.5 and γ = 2) (see Table 1). Figure 2 shows 10 simulated trajectories for x γ (t) (γ = 1 γ = 1.5 and γ = 2), taking as the values for α, β and σ those obtained by maximum likelihood estimation (see Table 2). For each trajectory, 2901 data are generated by considering h = 0.01, and initial value x 1 = 0.99.    Table 3. Table 3. Starting values used in the simulation and estimation of the parameters. The variation of the mean and standard error of the estimators is studied, taking into account how N and h change. The results are shown in Table 4. The next step is to study the evolution of the mean and the standard error of the estimators with respect to the variation in the number N and in h. The results of this study are shown in Table 4.
The calculations have been made using the Mathematica program, in which a program has been implemented.

Conclusions
This article presents a study of the Gamma Power Stochastic Gompertz Diffusion Process (γ-PSGDP), including all its probabilistic properties and the corresponding statistical inference. As a particular case in the limit comparison test, we also study the Gamma Power Stochastic Lognormal Diffusion Process (γ-PSLDP).
A simulation study was conducted, analysing different process trajectories.
In the future, it will be possible to apply these models to fit real data and to obtain goodness of fit results between the processes and the data. We will also study the possibility of defining all these processes in their non-homogeneous form, by introducing exogenous factors, and considering the use of numerical methods to obtain the estimates.
Author Contributions: All the authors have collaborated equally in the realization of this work, both in theoretical and applied developments. Similarly in the writing and review of it. All authors have read and agreed to the published version of the manuscript.

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript: