Inference with Non-Homogeneous Lognormal Diffusion Processes Conditioned on Nearest Neighbor

Abstract

In this work, we approach the forecast problem for a general non-homogeneous diffusion process over time with a different perspective from the classical one. We study the main characteristic functions as mean, mode, and α-quantiles conditioned on a future time, not conditioned on the past (as is normally the case), and we observe the specific formula in some interesting particular cases, such as Gompertz, logistic, or Bertalanffy diffusion processes, among others. This study aims to enhance classical inference methods when we need to impute data based on available information, past or future. We develop a simulation and obtain a dataset that is closer to reality, where there is no regularity in the number or timing of observations, to extend the traditional inference method. For such data, we propose using characteristic functions conditioned on the past or the future, depending on the closest point at which we aim to perform the imputation. The proposed inference procedure greatly reduces imputation errors in the simulated dataset.