CARMA processes as solutions of integral equations

A CARMA(p,q)(p,q) process is defined by suitable interpretation of the formal ppth order differential equation a(D)Yt=b(D)DLta(D)Yt=b(D)DLt, where LL is a two-sided Lévy process, a(z)a(z) and b(z)b(z) are polynomials of degrees pp and qq, respectively, with p>qp>q, and DD 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+rta(D)JpYt=b(D)Jp−1Lt+rt, where JJ, defined by Jft:=∫0tfsds, denotes the integration operator and rtrt is a suitable polynomial of degree at most p−1p−1. This equation has well defined solutions and provides a natural interpretation of the formal equation a(D)Yt=b(D)DLta(D)Yt=b(D)DLt.