Aspects of non‐causal and non‐invertible CARMA processes

Abstract

A CARMA(p, q) process Y is a strictly stationary solution Y of the pth-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 solutionIf 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.