1999/01/01 by Jean Bertoin · 9 citations
Mathematics · Economics, Econometrics and Finance · #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics #Stochastic processes and financial applications
paper · doi:10.1007/978-3-540-48115-7_1
openalex publication_date 1999/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A subordinator is an increasing process that has independent and homogeneous increments.Subordinators thus form one of the simplest family of random processes in continuous time.The purpose of this course is two-fold: First to expose salient features of the theory and second to present a variety of examples and applications.The theory mostly concerns the statistical and sample path properties.The applications we have in mind essentially follow from the connection between subordinators and regenerative sets, that can be thought of as the set of times when a Markov process visits some fixed point of the state space.Typically, this enables us to translate certain problems on a given Markov process in terms of some subordinator, and then to use general known results on the latter.Here is a sketch of the content.The first chapter introduces the basic notions and properties of subordinators, such as the Lévy-Khintchine formula, Itô's decomposition, renewal measures, ranges • • •, and the second presents the correspondence relating subordinators, regenerative sets, and local times and excursions of Markov processes, which is essential to the future applications.More advanced material in that field is developed in chapters 3-5, which concern respectively the asymptotic behaviour of last-passage times in connection with the Dynkin-Lamperti theorem, the smoothness of the local times (law of the iterated logarithm, modulus of continuity) and some geometric properties of regenerative sets including fractal dimensions and the study of the intersection with a given set.Applications are presented in chapters 6-9.First, we describe the law of the solution of the inviscid Burgers equation with Brownian initial velocity in terms of a subordinator, which enables us to investigate its statistical properties.Next, we study the closed subset of [0, ∞) that is left uncovered by open intervals sampled from a Poisson point process, following the ingenious approach of Fitzsimmons et al.Then, we turn our attention to two natural regenerative sets associated with a real-valued Lévy process: The set of passage times at a fixed state, and the set of times when a new maximum is achieved.Some applications of Bochner's subordination for Lévy processes are also given.Finally we investigate the class of subordinators that appears in connection with occupation times of a linear Brownian motion, or, equivalently, with the zero set of one-dimensional diffusions, by making use of M. G. Krein's spectral theory of vibrating strings.The choice of the examples discussed here is quite arbitrary; for instance, Marsalle [117] exposes further applications in the same vein, to increase times of stable processes, slow or fast points for local times, and the favorite site of a Brownian motion with drift.