Spectral representation of quasi-infinitely divisible processes

Riccardo Passeggeri

Published 2018-05-11Version 1

In the first part of this work we introduce quasi-infinitely divisible (QID) random measures, provide explicit examples and formulate a spectral representation of them. In the second part, we introduce QID stochastic integrals together with integrability conditions and continuity properties. In the last part, we introduces QID stochastic processes, whose definition is: a process $X$ is QID if and only if there exists two ID processes $Y$ and $Z$ such that $X+Y\stackrel{d}{=}Z$ with $Y$ independent of $X$. The class of QID processes is strictly larger than the class of ID processes. We present a spectral representation of discrete parameters QID processes. This work extends the results of the celebrated 1989 paper by Rajput and Rosinski to the QID framework.