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Continuous-time ARMA processes. (English) Zbl 1011.62088
Shanbhag, D. N. (ed.) et al., Stochastic processes: Theory and methods. Amsterdam: North-Holland/ Elsevier. Handb. Stat. 19, 249-276 (2001).
From the introduction: Continuous-time autoregressive (CAR) processes have been of interest to physicists and engineers for many years. In the last ten years there has been a resurgence of interest in continuous-time processes partly as a result of the very successful application of stochastic differential equation models to problems in finance, exemplified by the derivation of the Black-Scholes option-pricing formula and its generalizations. Continuous-time models have also been utilized very successfully for the modelling of irregularly-spaced data. At the same time there has been an increasing realization that nonlinear time series models provide much better representations of many empirically observed time series than linear models. The threshold ARMA models have been particularly successful in representing a wide variety of data sets, and the ARCH and GARCH models have had great success in the modelling of financial data.
In this paper we discuss continuous-time ARMA models, their basic properties, their relationship with discrete-time ARMA models, inference based on observations made at discrete times and nonlinear processes which include continuous-time analogues of H. Tong’s threshold ARMA models [see “Threshold models in nonlinear time series analysis.” (1983; Zbl 0527.62083)].
For the entire collection see [Zbl 0961.60001].

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)