A new stochastic Gompertz diffusion process with threshold parameter: Computational aspects and applications

https://doi.org/10.1016/j.amc.2006.05.099Get rights and content

Abstract

In this paper we propose a new homogeneous stochastic Gompertz diffusion model with a threshold parameter. This can be considered an extension of the homogeneous three parameter Gompertz process with the addition of a fourth parameter. From the corresponding Kolmogorov equations and Ito’s stochastic differential equations, we obtain the transition probability density function and the moments of this process (specifically, the trend functions). The parameters are estimated by considering discrete sampling of the sample path of the model and by using maximum likelihood methodology. Estimation of the threshold parameter requires us to solve a non-linear equation, which is achieved by the Newton–Raphson method. Simulated model data are considered and the methodology in question is applied to estimate the parameters; the values obtained are compared with those used in the simulation. Finally, the model is applied to model the evolution of the trend of the dynamic variable “average monthly salary cost”, for all sectors and broken down (construction, industry, services) in Spain, for the period (1985–2005).

Introduction

In recent decades, the notion of the stochastic diffusion process, defined by means of stochastic differential equations (SDE) or by the Kolmogorov equation, has been used in many fields, including economics, physics, engineering, cybernetics, environmetrics and biology.

The problem of estimating the parameters of the drift coefficient in these models has received considerable attention recently, especially in situations in which the process is observed continuously. The statistical inference is usually based on approximating maximum likelihood methodology. An extensive review of this theory can be found in Prakasa Rao [18], and related new work has been done by Bibby and Sorensen [1], Kloeden et al. [16], Singer [19] and others.

A wide variety of stochastic diffusion processes can be found in the literature, both general and specific. One such process is the stochastic Gompertz diffusion process (SGDP). From the point of view of stochastic differential equations, the homogeneous SGDP was introduced by Ricciardi [21] in a theoretical form, and subsequently applied by Ferrante et al. [4] (growth of cancer cells) and by Gutiérrez et al. [12] (consumption of natural gas in Spain). From the perspective of the Kolmogorov equations, the model was defined by Nafidi [17] in a general form, and later applied by Gutiérrez et al. [10] in a study of the stock of motor vehicles in Spain. The non-homogeneous form of the process (with exogenous factors) has been addressed by Nafidi [17] in a very general context. Later, Gutiérrez et al. [13], [15] studied the case in which only the growth rate in the drift is affected by exogenous factors in a linear way, and applied this both to the growth in the price of new housing in Spain and to the consumption of electricity in Morocco. Finally, Ferrante et al. [5] considered a non-homogeneous version in which the growth rate is the sum of two exponential functions that are exogenous factors.

As well as these non-homogeneous extensions of the SGDP, it would be useful to possess other extensions for real situations that might arise in various scientific fields. Frank [2] and Frank et al. [3] introduced a Gompertz diffusion process with delay, which was studied on the basis of the generalized Fokker–Planck equations with delays (given by Guillouzic et al. [23]). The present Gompertz model is used in the context of the stochastic system with delays.

With respect to introducing delay into the Gompertz diffusion process, another possibility might be to consider a threshold parameter. Thus, the model described in this paper is an extension of the homogeneous three-parameter SGDP model that is obtained by incorporating a threshold parameter in an analogous way to the procedure for homogeneous lognormal diffusion with two parameters when the threshold parameter is incorporated (see [11], [20]).

This parameter influences the dynamic variable under study, as well as its trend functions, and so we can obtain a better fit of the SGDP model to certain real phenomena that naturally present a threshold value in their behaviour pattern, i.e. a non-null minimum value from which the process trajectories evolve in time.

It is a known fact that the transition probability density function (TPDF) of a diffusion process, in general, cannot be expressed in closed form. Fortunately, for the process we propose, this function can be obtained, and thanks to its type (established as the density of a lognormal distribution), it offers the possibility of developing an inferential methodology based on the use of a discrete sampling of its trajectories, which is different from what was studied, for example, by [6], [12], [14], and which enables us to estimate its trend functions (both conditioned and non-conditioned), which are the necessary tools for fitting and predicting real phenomena.

This paper is organised, henceforth, as follows: in the second section the proposed model is defined on the basis of the corresponding Kolmogorov equations and on that of Ito’s stochastic differential equation, to obtain the TPDF and its moments. Subsequently, we estimate the parameters of the model by means of the maximum likelihood model, using discrete sampling of the process. Estimation of the threshold parameter requires us to resolve a non-linear equation, which is done by means of the Newton Raphson method. Section 3 contains a simulation of the exact solution of Ito’s SDE which characterises the process, thus illustrating the methodology by the simulation of its trajectories with respect to the theoretical trend function. The simulated process data are used to estimate the parameters of the model using the proposed methodology and these are compared with the true values used for the simulation. In Section 4, we describe an application of the process studied to model the evolution of the trend of the dynamic variable “average monthly salary cost”, both overall and by sectors (construction, industry, services) in Spain, using the data base for the period (1985–2005).

Section snippets

The stochastic diffusion Gompertz with threshold parameters

The one-dimensional Gompertz diffusion process with threshold parameter γ can be defined as a Markov process {Xt, t  [t0, T], t0 > 0}, taking values on ]γ, + ∞[ with almost certainly continuous paths and the infinitesimal momentsA(x)=α(x-γ)-β(x-γ)log(x-γ),B(x)=σ2(x-γ)2,where αR, β, σ and γ are positive real numbers (to be estimated). In the growth population, α is the intrinsic growth rate; β is the deceleration factor and σ is the diffusion coefficient volatility.

Let f(y, t/x, s) be the transition

Simulation studies

The trajectory of the model can be obtained by simulation of the exact solution of the equation (2). This solution can be obtained by means of Itô’s formula applied to the transform eβtlog(Xt  γ), from which we obtain the following equation:d[eβtlog(Xt-γ)]=(α-σ2/2)e-βtdt+σe-βtdWt.

By simplifying and integrating, the solution of the equation (2) leads us toXt=γ+explog(xs-γ)e-β(t-s)+α-σ2/2β(1-e-β(t-s))×expσste-β(t-τ)dWτ.From this explicit solution, we obtain the simulated trajectories of the

Application to reals data

In this application, we examine the average salary cost per worker and per month, by sectors (all activities, construction, services and industry) for the period 1986–2005. The following time-dependent random variables are considered:

  • X1(t): average salary cost per worker and per month, for all activities.

  • X2(t): average salary cost per worker and per month, for the construction sector.

  • X3(t): average salary cost per worker and per month, for the industrial sector.

  • X4(t): average salary cost per

Discussions and conclusions

  • From a theoretical standpoint, the main conclusion to be drawn from the present study is that it is possible to utilise a Gompertz homogeneous diffusion process that contains a γ > 0 parameter in such a way that the process is defined in ]γ, ∞[, and that this extends to the Gompertz process the idea of the existence of a “threshold parameter”, which has previously been considered both for probabilistic distributions (for example, in the distributions of extreme values in reliability studies) and

Acknowledgements

This work was supported partially by research project MTM 2005-09209, Ministerio de Educación y Ciencia, Spain.

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