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2 results

  • 11 - Applications

  • Denis Denisov, University of Manchester, Dmitry Korshunov, Lancaster University, Vitali Wachtel, Universität Bielefeld, Germany
  • Book:
    Markov Chains with Asymptotically Zero Drift
    Published online:
    24 April 2025
    Print publication:
    08 May 2025, pp 339-399

  • Summary

    The main goal of Chapter 11 is to demonstrate how the theory developed in the previous chapters can be used in the study of various Markov models that give rise to Markov chains with asymptotically zero drift. Some of those models are popular in stochastic modelling: random walks conditioned to stay positive, state-dependent branching processes or branching processes with migration, stochastic difference equations. In contrast with the general approach discussed here, the methods available in the literature for investigation of these models are mostly model tailored. We also introduce some new models to which our approach is applicable. For example, we introduce a risk process with surplus-dependent premium rate, which converges to the critical threshold in the nett-profit condition. Furthermore, we introduce a new class of branching processes with migration and with state-dependent offspring distributions.

Markov Chains with Asymptotically Zero Drift
  • Lamperti's Problem
  • Denis Denisov, Dmitry Korshunov, Vitali Wachtel
  • Published online:
    24 April 2025
    Print publication:
    08 May 2025
  • 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.