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 solution as $Y_t={\int }_{-\infty }^{\infty }g(t-u)\text{d}{L}_u$. 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.

中文翻译：

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非因果和不可逆 CARMA 过程的各个方面

甲CARMA（p，  q）工艺ý是一个严格的固定溶液ÿ所述的p阶正式随机微分方程一（d）ý_吨 =  b（d）DL_吨，其中大号是双面Levy过程，一个( z ) 和b ( z ) 分别是p 次和q 次的多项式，其中p  >  q和D表示关于t 的微分。使用定义方程的状态空间公式，Brockwell 和 Lindner (2009, Stochastic Processes and their Applications 119, 2660–2681) 给出了L、a ( z ) 和b ( z ) 存在性和唯一性的充分必要条件并在解的表示中指定核g为${是}_{吨}={\int }_{-\infty }^{\infty }G(吨-你)\text{d}{升}_{你}$. 如果a ( z )的零点都具有严格的负实部，则称Y是L的因果函数（或简单的因果函数），因为那时Y _t可以表示为L _s的增量，s  ≤  t，并且如果零点b（ż）都具有严格为负的实部的处理被认为是因为，则增量可逆大号_小号，小号 ≤ 吨，可以以的形式表示ÿ_小号，小号 ≤ 吨. 在本文中，我们关注 CARMA 过程的属性，对于这些属性，a和b上的这些条件不一定成立。