Statistics and Its Interface

Volume 15 (2022)

Number 1

Stochastic diffusion process based on Goel–Okumoto curve: statistical inference and application to real data

Pages: 63 – 71



Ahmed Nafidi (Hassan First University of Settat, Berrechid, Morocco)

Oussama Rida (Hassan First University of Settat, Berrechid, Morocco)

Meriem Bahij (Hassan First University of Settat, Berrechid, Morocco)

Boujemaa Achchab (Hassan First University of Settat, Berrechid, Morocco)


In this paper we study a new stochastic diffusion process based on the Goel-Okumoto curve. Such a process can be considered as an extension of the nonhomogeneous lognormal diffusion process. From the corresponding Itô’s stochastic differential equation (SDE), firstly we establish the probabilistic characteristics of the studied process, such as the solution to the SDE, the probability transition density function and their distribution, the moments function, in particular the conditional and non-conditional trend functions. Secondly, we treat the parameters estimation problem by using the maximum likelihood method in basis of the discrete sampling, thus we obtain nonlinear equations that can be solved by numerical methods. Finally, the proposed model is applied to the data of the broad money (% GDP) of Morocco.


Goel–Okumoto curve, diffusion process, likelihood estimation, forecast accuracy, broad money

2010 Mathematics Subject Classification

Primary 65C30. Secondary 60H30.

Received 19 January 2021

Accepted 30 March 2021

Published 11 August 2021