Three-parameter stochastic lognormal diffusion model: statistical computation and simulating annealing – application to real case

Abstract

In this paper, we propose a new study of a stochastic lognormal diffusion process (SLDP), with three parameters, which can be considered as an extension of the bi-parametric lognormal process with the addition of a threshold parameter. From the Kolmogorov equation, we obtain the probability density function and the moments of this process. The statistical inference of the parameter is studied by considering discrete sampling of the sample paths of the model and then using the maximum likelihood (ML) method. The estimation of the threshold parameter requires the solution of a nonlinear equation. To do so, we propose two methods: the classical Newton–Raphson (NR) method and one based on simulated annealing (SA). This methodology is applied to an example with simulated data corresponding to the process with known parameters. From this, we obtain the estimators of the parameters by both methods (NR and SA). Finally, the methodology studied is applied to a real case concerning the mean age of males in Spain at the date of their first wedding.