Aspects of non‐causal and non‐invertible CARMA processes
研究了当特征多项式零点不满足负实部条件时，连续时间自回归移动平均（CARMA）过程的性质，包括存在性、唯一性和表示核。
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 . 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.