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Summary

In Chapter 2 we introduce a classification of Markov chains with asymptotically zero drift, which relies on relations between the drift and the second moment of jumps, with many improvements on the results known in the literature. Additional assumptions are expressed in terms of truncated moments of higher orders and tail probabilities of jumps. Another, more important, contrast with previous results on recurrence/transience is the fact that we do not use concrete Lyapunov test functions (quadratic or similar). Instead, we construct an abstract Lyapunov function which is motivated by the harmonic function of a diffusion process with the same drift and diffusion coefficient.

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.

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2 - Lyapunov Functions and Classification of Markov Chains

Summary

Markov Chains with Asymptotically Zero Drift

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2 results

2 - Lyapunov Functions and Classification of Markov Chains

Summary

Markov Chains with Asymptotically Zero Drift