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STS517 Andreas Basse-O’Connor

Sufficient conditions for càdlàg sample paths of
stable superpositions of Ornstein–Uhlenbeck
processes
Andreas Basse-O’Connor
Department of Mathematics, Aarhus University, Aarhus, Denmark

Abstract
In this paper we derive sufficient conditions for stable superpositions of
Ornstein–Uhlenbeck processes to have cádlág sample paths with probability
one.

Keywords
SupOU processes; cádlág; sample path properties; stable processes

1. Introduction
The aim of the present paper is to obtain regularity results for stable
superpositions of Ornstein–Uhlenbeck (supOU) processes. These processes
are, in general, non-Markovian and non-semi martingale and have a
complicated dependence and path structure. The two main path regularities
used in the context of stochastic processes are continuity and cá dlá g. The
acronym cádlág comes from the French à droite, limite à gauche, which means
right continuous with left limits, and in particular, continuity implies cádlág.
Since stable supOU processes never have continuous sample paths (except in
the Gaussian case), it is natural to focus on when they have cádlág sample
paths, which is the content of Theorem 3.1.
In the following we will introduce supOU processes. A stable Ornstein–
Uhlenbeck (OU) process = {(): ≥ 0} is the solution to the stochastic
differential equation

where = {() ∶ ≥ 0} is a stable L´evy process and > 0. An Ornstein–
Uhlenbeck process is both Markovian, a semi martingale and has càdlàg
sample paths. Furthermore, the stationary solution to (1.1), which always exists,
can be represented on the form

where is extended to a two-sided L´evy process {(): ∈ }. Going beyond
Ornstein–Uhlenbeck processes, convex combinations of independent
Ornstein–Uhlenbeck processes (superpositions) are often used as models for

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