Abstract
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 CO\(_2\) and N\(_2\)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.
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References
Kloeden PE, Platen E, Kloeden PE, Platen E (1992) Stochastic differential equations. Springer, Berlin
Gutiérrez R, Gutiérrez-Sánchez R, Nafidi A (2008) Trend analysis using nonhomogeneous stochastic diffusion processes. Emission of co 2; kyoto protocol in spain. Stochast Environ Res Risk Assess 22:57–66
Nafidi A, Makroz I, Achchab B, Gutiérrez-Sánchez R (2023) Stochastic pareto diffusion process: statistical analysis and computational issues. Simulation and application. Moroccan J Pure Appl Anal 9(1):127–140
Nafidi A, El Azri A, Sánchez RG (2022) The stochastic modified lundqvist-korf diffusion process: statistical and computational aspects and application to modeling of the co 2 emission in Morocco. Stochast Environ Res Risk Assess 36(4):1163–1176
Aït-Sahalia Y (2008) Closed-form likelihood expansions for multivariate diffusions. Ann Stat 36(1):906–937
Prakasa Rao BLS (1999) Statistical inference for diffusion type processes. Arnold London
Gutiérrez R, Angulo J, González A, Pérez R (1991) Inference in lognormal multidimensional diffusion processes with exogenous factors: application to modelling in economics. Appl Stochast Models Data Anal 7(4):295–316
Frank T (2002) Multivariate markov processes for stochastic systems with delays: application to the stochastic gompertz model with delay. Phys Rev E 66(1):011914
Gutiérrez-Jáimez R, Gutiérrez-Sánchez R, Nafidi A, Ramos-Ábalos EM (2014) A bivariate stochastic gamma diffusion model: statistical inference and application to the joint modelling of the gross domestic product and co2 emissions in spain. Stochast Environ Res Risk Assess 28(5):1125–1134
Varughese MM, Pienaar EAD (2013) Statistical inference for a multivariate diffusion model of an ecological time series. Ecosphere 4(8):1–14
Jaimez RG, Carmona AG, Ruiz FT (1997) Algorithm as 309: estimation in multivariate log-normal diffusion processes with exogenous factors. Appl Stat, pp 140–146
Vasicek O (1977) An equilibrium characterization of the term structure. J Financial Econ 5(2):177–188
Cox JC, Ross SA (1976) The valuation of options for alternative stochastic processes. J Financial Econ 3(1–2):145–166
Hull J, White A (2001) The general hull-white model and supercalibration. Financ Anal J 57(6):34–43
Gutiérrez R, Gutiérrez-Sánchez R, Nafidi A, Pascual A (2012) Detection, modelling and estimation of non-linear trends by using a non-homogeneous vasicek stochastic diffusion. application to co2 emissions in morocco. Stochast Environ Res Risk Assess 26(4):533–543
Albano G, Rocca ML, Perna C (2021) A comparison among alternative parameters estimators in the vasicek process: a small sample analysis. In: Mathematical and Statistical Methods for Actuarial Sciences and Finance: eMAF2020, pp 1–6. Springer
Narmontas M, Rupšys P, Petrauskas E (2020) Models for tree taper form: The gompertz and vasicek diffusion processes framework. Symmetry 12(1):80
Arnold L (1974) Stochastic differential equations. New York
Magnus JR, Neudecker H (2019) Matrix differential calculus with applications in statistics and econometrics. John Wiley & Sons, London
Zehna PW (1966) Invariance of maximum likelihood estimators. Ann Math Stat 37(3):744
Stocker TF, Qin D, Plattner G-K, Tignor MM, Allen SK, Boschung J, Nauels A, Xia Y, Bex, V, Midgley PM (2014) Climate change 2013: the physical science basis. In: Contribution of working group i to the fifth assessment report of ipcc the intergovernmental panel on climate change
Montzka SA, Dlugokencky EJ, Butler JH (2011) Non-co2 greenhouse gases and climate change. Nature 476(7358):43–50
IPCC A et al (2013) Climate change 2013: the physical science basis. In: Contribution of working group I to the fifth assessment report of the intergovernmental panel on climate change, 1535
Netz B, Davidson OR, Bosch PR, Dave R, Meyer LA et al (2007) Contribution of working group iii to the fourth assessment report of the intergovernmental panel on climate change. summary for policymakers. Climate change 2007: Mitigation
Misra A, Verma M (2013) A mathematical model to study the dynamics of carbon dioxide gas in the atmosphere. Appl Math Comput 219(16):8595–8609
Sephton PS (2022) Further evidence of mean reversion in co2 emissions. World Develop Sustain 1:100021
Ritchie H, Roser M, Rosado P (2020) Co2 and greenhouse gas emissions. Our World in Data. https://ourworldindata.org/co2-and-other-greenhouse-gas-emissions
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Nafidi, A., Makroz, I., Gutiérrez Sánchez, R. et al. Multivariate stochastic Vasicek diffusion process: computational estimation and application to the analysis of \(CO_2\) and \(N_2O\) concentrations. Stoch Environ Res Risk Assess 38, 2581–2590 (2024). https://doi.org/10.1007/s00477-024-02699-y
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DOI: https://doi.org/10.1007/s00477-024-02699-y