Recent results in the theory and applications of CARMA processes

Just as ARMA processes play a central role in the representation of stationary time series with discrete time parameter, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$(Y_n)_{n\in \mathbb {Z}}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>Y</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">Z</mi> </mrow> </msub> </math> </EquationSource> </InlineEquation>, CARMA processes play an analogous role in the representation of stationary time series with continuous time parameter, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$(Y(t))_{t\in \mathbb {R}}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>t</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </msub> </math> </EquationSource> </InlineEquation>. 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 (<CitationRef CitationID="CR18">2001a</CitationRef>, 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