Aspects of non‐causal and non‐invertible CARMA processes. (6th May 2021)

• Record Type: Journal Article
• Authors: Brockwell, Peter J.; Lindner, Alexander
• Other Names: Bradley Richard C. guestEditor.; Davis Richard A. guestEditor.; Politis Dimitris N. guestEditor.
• Is Part Of: Journal of time series analysis. Volume 42:Number 5/6(2021)
• Page Start: 777
• Page End: 790
• Publication Date: 2021-05-06
• Subjects: Lévy process -- CARMA process -- causality -- invertibility. MOS subject classification: 60G10 -- 60G51 -- 60G; Time-series analysis -- Periodicals; 519.232
• DOI: 10.1111/jtsa.12589
• Languages: English
• ISSNs: 0143-9782

Abstract

Abstract : A CARMA( p, q ) process Y is a strictly stationary solution Y of the p th‐order formal stochastic differential equation a ( D ) Y t = b ( D ) DL t, where L is a two‐sided Lévy process, a ( z ) and b ( z ) are polynomials of degrees p and q respectively, with p > q, and D denotes differentiation with respect to t . Using a state‐space formulation of the defining equation, Brockwell and Lindner (2009, Stochastic Processes and their Applications 119, 2660–2681) gave necessary and sufficient conditions on L, a ( z ) and b ( z ) for the existence and uniqueness of such a stationary solution and specified the kernel g in the representation of the solution as Y t = ∫ − ∞ ∞ g ( t − u ) d L u . If the zeros of a ( z ) all have strictly negative real parts, Y is said to be a causal function of L (or simply causal) since then Y t can be expressed in terms of the increments of L s, s ≤ t, and if the zeros of b ( z ) all have strictly negative real parts the process is said to be invertible since then the increments of L s, s ≤ t, can be expressed in terms of Y s, s ≤ t . In this article we are concerned with properties of CARMA processes for which these conditions on a and b do not necessarily hold.