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This chapter contains an account of contemporary Lévy theory, namely the theory of infinite divisibility, Lévy processes, Lévy bases, and so forth, which has been a very active area of research over the past decade. It surveys a few aspects and ramifications of these developments, the choice of topics being determined by its author’s interests rather than being intended to form a comprehensive overview. After an exposition of classical infinite divisibility and related topics, there is an account of Lévy processes and bases, and of Ornstein-Uhlenbeck processes. A general class of tempospatial models is sketched, and time change, applications to finance and turbulence, and the notion of an upsilon mapping are discussed. The chapter closes with an overview of the links between quantum stochastics and Lévy theory.

This chapter contains a survey of classical probability theory and stochastic processes. It starts with a description of the fundamental concepts of probability space and Kolmogorov axioms. These concepts are then used to define random variables and stochastic processes. The mathematical formulation of the special class of Markov processes through classical master equations is given, including deterministic processes (Liouville equation), jump processes (Pauli master equation), and diffusion processes (Fokker–Planck equation). Special stochastic processes which play an important role in the developments of the following chapters, such as piecewise deterministic processes and Lévy processes, are described in detail together with their basic physical properties and various mathematical formulations in terms of master equations, path integral representation, and stochastic differential equations.

We study the classic problem of choosing a prior distribution for a location parameter β = (β ₁,…, β _p ) as p grows large. First, we study the standard “global‐local shrinkage” approach, based on scale mixtures of normals. Two theorems are presented which characterize certain desirable properties of shrinkage priors for sparse problems. Next, we review some recent results showing how Lévy processes can be used to generate infinite‐dimensional versions of standard normal scale‐mixture priors, along with new priors that have yet to be seriously studied in the literature. This approach provides an intuitive framework both for generating new regularization penalties and shrinkage rules, and for performing asymptotic analysis on existing models.

This chapter presents a quick review of the theory of semimartingales, which are processes for which statistical methods are considered in this book. Topics covered include diffusions, Lévy processes, Itô semimartingales, and processes with conditionally independent increments.

This chapter examines Lévy processes (LPs). These processes can be thought of as random walks in continuous time, having independent and stationary increments, with the assumption that the characteristic function of the state of an LP at time 1 satisfies a certain “Standard Condition” (SC). Exponential Lévy processes (ELPs) in particular are quite attractive for modeling many phenomena. The chapter begins by explaining LPs in more detail, introducing the basic notions as well as providing examples of Lévy processes. It then uses a certain variant of the Poisson Summation Formula to arrive at convergence results for the expectations of certain normalized functionals of the significand of an ELP and obtain, using Azuma's inequality for martingales, large deviation results for these functionals. From here, the chapter calculates the a.s. (almost surely) convergence of normalized functionals and discusses further related conditions and theorems.

This chapter deals with a presentation of continuous measurement theory on the basis of the microscopic equations of quantum electrodynamics. The emphasis lies on the derivation of the quantum operations and the corresponding stochastic processes for various detection schemes directly from the Hamiltonian, which describes the interaction of the matter degrees of freedom with the quantized electromagnetic field. The chapter treats many examples and applications to atomic physics and quantum optics, such as dark state resonances and laser cooling of atoms. This example illustrates the interplay between incoherent processes and quantum interference effects which leads to the emergence of Lévy-type distributions for the atomic waiting time. Finally, dissipative phenomena in the dynamics of open quantum systems in strong driving fields are studied, for which appropriate master equations and stochastic wave function methods can be derived by employing a representation in terms of Floquet states.

This chapter begins with a simple general framework for inference in the presence of α‎-stable processes, where the stable processes are represented as conditionally Gaussian distributions, relying on (exact) series representations of the stable laws and the corresponding stochastic integrations in terms of infinite summations of random Poisson process arrival times. Inference can therefore be carried out using techniques including auxiliary variables, Rao-Blackwellized particle filtering, and Markov chain Monte Carlo. The Poisson series representation is further enhanced by introducing an approximation of the series residual terms based on exact moment calculations. Extensions to the discrete-time asymmetric stable case and to continuous-time are

This chapter starts with a brief reminder about a number of concepts and results which pertain to classical statistical models, without specific reference to stochastic processes. It then introduces a general notion of identifiability for a parameter, in a semi-parametric setting. A parameter can be a number (or a vector), as in classical statistics; it can also be a random variable, such as the integrated volatility. The analysis is first conducted for Lévy processes, because in this case parameters are naturally non-random, and then extended to the more general situation of semimartingales. It also considers the problem of testing a hypothesis which is “random,” such as testing whether a discretely observed path is continuous or discontinuous: the null and alternative are not the usual disjoint subsets of a parameter space, but rather two disjoint subsets of the sample space, which leads to an ad hoc definition of the level, or asymptotic level, of a test in such a context. Finally, the chapter returns to the question of efficient estimation of a parameter, which is mainly analyzed from the viewpoint of “Fisher efficiency.”