On infinitely divisible semimartingales

TL;DR

This paper extends Stricker's theorem from Gaussian to a broad class of infinitely divisible processes, providing a unique decomposition into independent increment and finite variation parts, and characterizing semimartingales in this context.

Contribution

It proves a new decomposition theorem for infinitely divisible semimartingales, generalizing Stricker's theorem beyond Gaussian processes and characterizing semimartingales via path properties.

Findings

The class of infinitely divisible semimartingales is very large, and the natural analog of Stricker's theorem generally fails.

A unique decomposition exists for infinitely divisible semimartingales relative to the filtration generated by a random measure.

The result applies to many processes of interest, including linear fractional, mixed moving averages, and supOU processes.

Abstract

Stricker's theorem states that a Gaussian process is a semimartingale in its natural filtration if and only if it is the sum of an independent increment Gaussian process and a Gaussian process of finite variation, see [1983, Z. Wahrsch. Verw. Gebiete 64(3)]. We consider extensions of this result to non Gaussian infinitely divisible processes. First we show that the class of infinitely divisible semimartingales is so large that the natural analog of Stricker's theorem fails to hold. Then, as the main result, we prove that an infinitely divisible semimartingale relative to the filtration generated by a random measure admits a unique decomposition into an independent increment process and an infinitely divisible process of finite variation. Consequently, the natural analog of Stricker's theorem holds for all strictly representable processes (as defined in this paper). Since Gaussian processes are strictly representable due to Hida's multiplicity theorem, the classical Stricker's theorem follows from our result. Another consequence is that the question when an infinitely divisible process is a semimartingale can often be reduced to a path property, when a certain associated infinitely divisible process is of finite variation. This gives the key to characterize the semimartingale property for many processes of interest. Along these lines, we characterize semimartingales within a large class of stationary increment infinitely divisible processes; this class includes many infinitely divisible processes, including linear fractional processes, mixed moving averages, and supOU processes, as particular cases. The proof of the main theorem relies on series representations of jumps of cadlag infinitely divisible processes given in Basse-O'Connor and Rosinski [2013, Ann. Probab. 41(6)] combined with techniques of stochastic analysis.