CARMA processes as solutions of integral equations

Peter J. Brockwell and Alexander Lindner

Statistics & Probability Letters, 2015, vol. 107, issue C, 221-227

Abstract: A CARMA(p,q) process is defined by suitable interpretation of the formal pth order differential equation a(D)Yt=b(D)DLt, 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 the differentiation operator. Since derivatives of Lévy processes do not exist in the usual sense, the rigorous definition of a CARMA process is based on a corresponding state space equation. In this note, we show that the state space definition is also equivalent to the integral equation a(D)JpYt=b(D)Jp−1Lt+rt, where J, defined by Jft:=∫0tfsds, denotes the integration operator and rt is a suitable polynomial of degree at most p−1. This equation has well defined solutions and provides a natural interpretation of the formal equation a(D)Yt=b(D)DLt.

Keywords: CARMA process; Continuous time autoregressive moving average process; Differential equation; Integral equation (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1016/j.spl.2015.08.026

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