# Recent results in the theory and applications of CARMA processes

Listed:
• P. Brockwell

()

## Abstract

Just as ARMA processes play a central role in the representation of stationary time series with discrete time parameter, $$(Y_n)_{n\in \mathbb {Z}}$$ ( Y n ) n ∈ Z , CARMA processes play an analogous role in the representation of stationary time series with continuous time parameter, $$(Y(t))_{t\in \mathbb {R}}$$ ( Y ( t ) ) t ∈ R . Lévy-driven CARMA processes permit the modelling of heavy-tailed and asymmetric time series and incorporate both distributional and sample-path information. In this article we provide a review of the basic theory and applications, emphasizing developments which have occurred since the earlier review in Brockwell ( 2001a , In D. N. Shanbhag and C. R. Rao (Eds.), Handbook of Statistics 19; Stochastic Processes: Theory and Methods (pp. 249–276), Amsterdam: Elsevier). Copyright The Institute of Statistical Mathematics, Tokyo 2014

## Suggested Citation

• 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.
• Handle: RePEc:spr:aistmt:v:66:y:2014:i:4:p:647-685
DOI: 10.1007/s10463-014-0468-7
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File URL: http://hdl.handle.net/10.1007/s10463-014-0468-7

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## Citations

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Cited by:

1. Nielsen, Mikkel Slot, 2020. "On non-stationary solutions to MSDDEs: Representations and the cointegration space," Stochastic Processes and their Applications, Elsevier, vol. 130(5), pages 3154-3173.
2. Chambers, MJ & McCrorie, JR & Thornton, MA, 2017. "Continuous Time Modelling Based on an Exact Discrete Time Representation," Economics Discussion Papers 20497, University of Essex, Department of Economics.
3. Valerie Girardin & Rachid Senoussi, 2020. "Filling the gap between Continuous and Discrete Time Dynamics of Autoregressive Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 41(4), pages 590-602, July.
4. Sikora, Grzegorz & Michalak, Anna & Bielak, Łukasz & Miśta, Paweł & Wyłomańska, Agnieszka, 2019. "Stochastic modeling of currency exchange rates with novel validation techniques," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 523(C), pages 1202-1215.
5. Bai, Shuyang & Ginovyan, Mamikon S. & Taqqu, Murad S., 2016. "Limit theorems for quadratic forms of Lévy-driven continuous-time linear processes," Stochastic Processes and their Applications, Elsevier, vol. 126(4), pages 1036-1065.
6. Müller, Gernot & Seibert, Armin, 2019. "Bayesian estimation of stable CARMA spot models for electricity prices," Energy Economics, Elsevier, vol. 78(C), pages 267-277.
7. Peter J. Brockwell & Yasumasa Matsuda, 2015. "Levy-driven CARMA Random Fields on Rn," TERG Discussion Papers 339, Graduate School of Economics and Management, Tohoku University.
8. Peter J. Brockwell & Yasumasa Matsuda, 2017. "Continuous auto-regressive moving average random fields on ℝ-super-n," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(3), pages 833-857, June.
9. Basse-O’Connor, Andreas & Nielsen, Mikkel Slot & Pedersen, Jan & Rohde, Victor, 2019. "Multivariate stochastic delay differential equations and CAR representations of CARMA processes," Stochastic Processes and their Applications, Elsevier, vol. 129(10), pages 4119-4143.
10. Lingohr, Daniel & Müller, Gernot, 2019. "Stochastic modeling of intraday photovoltaic power generation," Energy Economics, Elsevier, vol. 81(C), pages 175-186.

### Keywords

Time series; Stationary process; CARMA process ; Sampled process; High-frequency sampling; Inference ; Prediction;

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