Markov Chains with Asymptotically Zero Drift
Markov Chains with Asymptotically Zero Drift
Lamperti's Problem
Denis Denisov, University of Manchester
Dmitry Korshunov, Lancaster University
Vitali Wachtel, Universität Bielefeld, Germany
Dmitry Korshunov, Lancaster University
Vitali Wachtel, Universität Bielefeld, Germany
June 2025
Hardback
9781009554220
£120.00
GBPHardback
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.
- Includes many novel elements and much of the material presents original research
- Builds on the classical topic of asymptotic analysis and classification of Markov chains
- Offers a high-precision alternative to classical Lyapunov functions
Product details
June 2025Hardback
9781009554220
432 pages
229 × 152 mm
Not yet published - available from June 2025
Table of Contents
- 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.
