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Weak convergence of stochastic integrals with respect to the state occupation measure of a Markov chain

Part of: Stochastic analysis Limit theorems

Published online by Cambridge University Press:  23 June 2021

H. M. Jansen
H. M. Jansen*
Affiliation:
Delft University of Technology
*
*Postal address: Delft Institute of Applied Mathematics, Delft University of Technology, Van Mourik Broekmanweg 6, 2628 XE Delft, the Netherlands. Email address: h.m.jansen@tudelft.nl
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Abstract

Our aim is to find sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure of a Markov chain. First, we study properties of the state indicator function and the state occupation measure of a Markov chain. In particular, we establish weak convergence of the state occupation measure under a scaling of the generator matrix. Then, relying on the connection between the state occupation measure and the Dynkin martingale, we provide sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure. We apply our results to derive diffusion limits for the Markov-modulated Erlang loss model and the regime-switching Cox–Ingersoll–Ross process.

Keywords

Markov chainstate occupation measurestochastic integraldiffusion limitMarkov modulationregime switching

MSC classification

Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60H05: Stochastic integrals
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 2 , June 2021 , pp. 372 - 393
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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