Markov Processes, Gaussian Processes, and Local Times

Abstract

This book was first published in 2006. Written by two of the foremost researchers in the field, this book studies the local times of Markov processes by employing isomorphism theorems that relate them to certain associated Gaussian processes. It builds to this material through self-contained but harmonized 'mini-courses' on the relevant ingredients, which assume only knowledge of measure-theoretic probability. The streamlined selection of topics creates an easy entrance for students and experts in related fields. The book starts by developing the fundamentals of Markov process theory and then of Gaussian process theory, including sample path properties. It then proceeds to more advanced results, bringing the reader to the heart of contemporary research. It presents the remarkable isomorphism theorems of Dynkin and Eisenbaum and then shows how they can be applied to obtain new properties of Markov processes by using well-established techniques in Gaussian process theory. This original, readable book will appeal to both researchers and advanced graduate students.

Estimating the Volatility Occupation Time via Regularized Laplace Inversion

“…As in Assumption A, we only need the dominance conditions to hold locally up to the localizing sequence T m . Assumption B1 holds if the occupation density of V t exists, which is the case for general semimartingale processes with nondegenerate diffusive component and large classes of Gaussian processes; see, for example, Geman and Horowitz (1980), Protter (2004), Marcus and Rosen (2006), Eisenbaum and Kaspi (2007) and references therein. Assumption B1 holds more generally under settings where F t (·) can be nondifferentiable (and even discontinuous) at random points, as long as these irregular points are located "diffusively" on the line, as formulated by the second part of Assumption B1.…”

Dynkin’s Isomorphism Theorem and the Stochastic Heat Equation

“…Before we do that we identify precisely the family operators L with which we are concerned, since this requires some careful development. We refer the reader to the recent monograph by Marcus and Rosen [10], which contains a wealth of information on Markov processes, local times, and their deep connections to Gaussian processes. Our notation for Markov processes is standard and follows [10] as well.…”

“…By mutual absolute continuity, and thanks to Dynkin's isomorphism theorem [10,Chapter 8], it follows that many of the local features of L · t and U (t , ·) are shared. This explains the aforementioned connections between (1.1) and local times in the case that L is the generator of a Lévy process.…”

“…It is easy to deduce from general theory [10, that if G is smoother than G * , then the continuity of x → G * (x) implies the continuity of x → G(x). Similar remarks can be made about Hölder continuity and existence of nontrivial p-variations, in case the latter properties hold and/or make sense.…”

“…We refer the reader to Chapter 8 of the book by Marcus and Rosen [10] for details on Dynkin's isomorphism theorem and its applications to the analysis of local properties of local times.…”