Multivariate stochastic Vasicek diffusion process: computational estimation and application to the analysis of CO2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CO_2$$\end{document} and N2O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_2O$$\end{document} concentrations

In this paper, we propose a new extension of the Vasicek model to the multivariate case and analyze its characteristics, including its probability density function, marginal trends, and correlation functions. We also conduct statistical estimation on the model, including likelihood parameter estimation and estimation of the marginal trend and correlation functions. To validate the effectiveness of our proposed model, we use simulation studies and examine the goodness of fit. Additionally, we apply our model to a bivariate case involving the analysis of CO2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_2$$\end{document} and N2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_2$$\end{document}O concentrations. We describe the data, fit the bivariate Vasicek stochastic diffusion model, and compare it with the univariate case. Finally, we summarize our findings and discuss the potential applications of our proposed model.