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.

中文翻译：

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基于 Goel-Okumoto 曲线的随机扩散过程：统计推断及其在真实数据中的应用

在本文中，我们研究了一种基于 Goel-Okumoto 曲线的新随机扩散过程。这样的过程可以看作是非齐次对数正态扩散过程的扩展。从相应的伊藤随机微分方程 (SDE)，首先我们建立了所研究过程的概率特征，例如 SDE 的解、概率转移密度函数及其分布、矩函数，特别是条件和非条件趋势函数。其次，在离散抽样的基础上，采用极大似然法处理参数估计问题，从而得到可以用数值方法求解的非线性方程。最后，将提出的模型应用于摩洛哥的广义货币（% GDP）数据。