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Rate of strong convergence to Markov-modulated Brownian motion

Part of: Stochastic processes Markov processes Limit theorems

Published online by Cambridge University Press:  18 January 2022

Giang T. Nguyen Open the ORCID record for Giang T. Nguyen [Opens in a new window]  and
Oscar Peralta Open the ORCID record for Oscar Peralta [Opens in a new window]
Giang T. Nguyen*
Affiliation:
The University of Adelaide
Oscar Peralta*
Affiliation:
The University of Adelaide
*
*Postal address: The University of Adelaide, School of Mathematical Sciences, SA 5005, Australia.
*Postal address: The University of Adelaide, School of Mathematical Sciences, SA 5005, Australia.
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Abstract

Latouche and Nguyen (2015b) constructed a sequence of stochastic fluid processes and showed that it converges weakly to a Markov-modulated Brownian motion (MMBM). Here, we construct a different sequence of stochastic fluid processes and show that it converges strongly to an MMBM. To the best of our knowledge, this is the first result on strong convergence to a Markov-modulated Brownian motion. Besides implying weak convergence, such a strong approximation constitutes a powerful tool for developing deep results for sophisticated models. Additionally, we prove that the rate of this almost sure convergence is $o(n^{-1/2} \log n)$ . When reduced to the special case of standard Brownian motion, our convergence rate is an improvement over that obtained by a different approximation in Gorostiza and Griego (1980), which is $o(n^{-1/2}(\log n)^{5/2})$ .

Keywords

Stochastic fluid modelMarkov-modulated Brownian motionstrong convergencefirst passage probability

MSC classification

Primary: 60G15: Gaussian processes
Secondary: 60F15: Strong theorems 60J27: Continuous-time Markov processes on discrete state spaces
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 1 , March 2022 , pp. 1 - 16
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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