Markov Chains with Asymptotically Zero Drift

Lamperti's Problem
Erscheinungsjahr 2025
ISBN: 978-1-009-55422-0
Verlag: Cambridge University Press

Buch, Englisch, Band 51, 428 Seiten

Reihe: New Mathematical Monographs

This text examines Markov chains whose drift tends to zero at infinity, a topic sometimes labelled as 'Lamperti's problem'. It can be considered a subcategory of random walks, which are helpful in studying stochastic models like branching processes and queueing systems. Drawing on Doob's h-transform and other tools, the authors present novel results and techniques, including a change-of-measure technique for near-critical Markov chains. The final chapter presents a range of applications where these special types of Markov chains occur naturally, featuring a new risk process with surplus-dependent premium rate. This will be a valuable resource for researchers and graduate students working in probability theory and stochastic processes.

Autoren/Hrsg.

Denisov, Denis

Korshunov, Dmitry

Wachtel, Vitali

Inhaltsverzeichnis

1. Introduction; 2. Lyapunov functions and classification of Markov chains; 3. Down-crossing probabilities for transient Markov chain; 4. Limit theorems for transient and null-recurrent Markov chains with drift proportional to 1/x; 5. Limit theorems for transient Markov chains with drift decreasing slower than 1/x; 6. Asymptotics for renewal measure for transient Markov chain via martingale approach; 7. Doob's h-transform: transition from recurrent to transient chain and vice versa; 8. Tail analysis for recurrent Markov chains with drift proportional to 1/x; 9. Tail analysis for positive recurrent Markov chains with drift going to zero slower than 1/x; 10. Markov chains with asymptotically non-zero drift in Cramér's case; 11. Applications.

Über Autor(innen)

Wachtel, Vitali
Vitali Wachtel is Professor for Mathematics at the University of Bielefeld. His research interests include, besides Markov chains, exit times and conditional distributions for multidimensional random walks, branching processes and large deviations.

Denisov, Denis
Denis Denisov is Reader in Probability in the Department of Mathematics at the University of Manchester. His research interests include multidimensional random walks, ordered and conditioned random walks, Markov chains and diffusion processes.

Korshunov, Dmitry
Dmitry Korshunov is Professor in the School of Mathematical Sciences at Lancaster University. He is an expert in stochastic processes, Markov chains, large deviation theorems, heavy tail phenomena, and limit theorems. He previously co-authored An Introduction to Heavy-Tailed and Subexponential Distributions (2011, 2013).