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Multivariate continuous-time autoregressive moving-average processes on cones

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  • Benth, Fred Espen
  • Karbach, Sven

Abstract

In this article we study multivariate continuous-time autoregressive moving-average (MCARMA) processes with values in convex cones. More specifically, we introduce matrix-valued MCARMA processes with Lévy noise and present necessary and sufficient conditions for processes from this class to be cone valued. We derive specific hands-on conditions in the following two cases: First, for classical MCARMA on Rd with values in the positive orthant Rd+. Second, for MCARMA processes on real square matrices assuming values in the cone of symmetric and positive semi-definite matrices. Both cases are relevant for applications and we give several examples of positivity ensuring parameter specifications. In addition to the above, we discuss the capability of positive semi-definite MCARMA processes to model the spot covariance process in multivariate stochastic volatility models. We justify the relevance of MCARMA based stochastic volatility models by an exemplary analysis of the second order moment structure of positive semi-definite well-balanced Ornstein–Uhlenbeck based models.

Suggested Citation

  • Benth, Fred Espen & Karbach, Sven, 2023. "Multivariate continuous-time autoregressive moving-average processes on cones," Stochastic Processes and their Applications, Elsevier, vol. 162(C), pages 299-337.
    • Handle: RePEc:eee:spapps:v:162:y:2023:i:c:p:299-337
    DOI: 10.1016/j.spa.2023.05.003
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    References listed on IDEAS

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    1. Brockwell, Peter J. & Davis, Richard A. & Yang, Yu, 2011. "Estimation for Non-Negative Lévy-Driven CARMA Processes," Journal of Business & Economic Statistics, American Statistical Association, vol. 29(2), pages 250-259.
    2. Chen, Rong & Xiao, Han & Yang, Dan, 2021. "Autoregressive models for matrix-valued time series," Journal of Econometrics, Elsevier, vol. 222(1), pages 539-560.
    3. Vicky Fasen, 2016. "Dependence Estimation for High-frequency Sampled Multivariate CARMA Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(1), pages 292-320, March.
    4. Cox, Sonja & Karbach, Sven & Khedher, Asma, 2022. "Affine pure-jump processes on positive Hilbert–Schmidt operators," Stochastic Processes and their Applications, Elsevier, vol. 151(C), pages 191-229.
    5. Christa Cuchiero & Martin Keller-Ressel & Eberhard Mayerhofer & Josef Teichmann, 2016. "Affine Processes on Symmetric Cones," Journal of Theoretical Probability, Springer, vol. 29(2), pages 359-422, June.
    6. Peter Brockwell & Alexander Lindner, 2013. "Integration of CARMA processes and spot volatility modelling," Journal of Time Series Analysis, Wiley Blackwell, vol. 34(2), pages 156-167, March.
    7. Peter J. Brockwell & Richard A. Davis & Yu Yang, 2011. "Estimation for Non-Negative Lévy-Driven CARMA Processes," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 29(2), pages 250-259, April.
    8. Marquardt, Tina & Stelzer, Robert, 2007. "Multivariate CARMA processes," Stochastic Processes and their Applications, Elsevier, vol. 117(1), pages 96-120, January.
    9. Brockwell, Peter J. & Schlemm, Eckhard, 2013. "Parametric estimation of the driving Lévy process of multivariate CARMA processes from discrete observations," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 217-251.
    10. Henghsiu Tsai & K. S. Chan, 2005. "A note on non‐negative continuous time processes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(4), pages 589-597, September.
    11. Fred E. Benth & Troels S. Christensen & Victor Rohde, 2021. "Multivariate continuous-time modeling of wind indexes and hedging of wind risk," Quantitative Finance, Taylor & Francis Journals, vol. 21(1), pages 165-183, January.
    12. P. Brockwell, 2014. "Recent results in the theory and applications of CARMA processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(4), pages 647-685, August.
    13. Marquardt, Tina, 2007. "Multivariate fractionally integrated CARMA processes," Journal of Multivariate Analysis, Elsevier, vol. 98(9), pages 1705-1725, October.
    14. Christa Cuchiero & Damir Filipovi'c & Eberhard Mayerhofer & Josef Teichmann, 2009. "Affine processes on positive semidefinite matrices," Papers 0910.0137, arXiv.org, revised Apr 2011.
    15. Peter J. Brockwell & Alexander Lindner, 2021. "Aspects of non‐causal and non‐invertible CARMA processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 42(5-6), pages 777-790, September.
    16. Todorov, Viktor & Tauchen, George, 2006. "Simulation Methods for Levy-Driven Continuous-Time Autoregressive Moving Average (CARMA) Stochastic Volatility Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 24, pages 455-469, October.
    17. Benth, Fred Espen & Saltyte Benth, Jurate, 2009. "Dynamic pricing of wind futures," Energy Economics, Elsevier, vol. 31(1), pages 16-24, January.
    18. Péter Kevei, 2018. "Asymptotic moving average representation of high-frequency sampled multivariate CARMA processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 70(2), pages 467-487, April.
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