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CARMA processes as solutions of integral equations. (English) Zbl 1356.60072

Summary: A CARMA\((p, q)\) process is defined by suitable interpretation of the formal \(p\)th order 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 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 \(a(D) J^p Y_t = b(D) J^{p - 1} L_t + r_t\), where \(J\), defined by \(Jf_t : = \int_0^tf_s\operatorname{d}s\), denotes the integration operator and \(r_t\) 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 \(a(D)Y_t = b(D)DL_t\).

MSC:

60G51 Processes with independent increments; Lévy processes
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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