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

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).