CARMA processes as solutions of integral equations.

A CARMA ( p , q ) process is defined by suitable interpretation of the formal p th order differential equation a ( D ) Y t = b ( D ) D L 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 J f t : = ∫ 0 t f s 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 ) D L t . [ABSTRACT FROM AUTHOR]