Markov chains with asymptotically zero drift : Lamperti's problem

Denisov, Denis and Korshunov, Dmitry and Wachtel, Vitali (2025) Markov chains with asymptotically zero drift : Lamperti's problem. New Mathematical Monographs . Cambridge University Press, Cambridge. ISBN 9781009554220

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Abstract

This text examines Markov chains whose drift tends to zero at infinity, a topic sometimes referred to as Lamperti’s problem. It can be considered a subcategory of random walks, which are helpful in studying stochastic models such as 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 chain 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.