Rate of Strong Convergence to Markov-modulated Brownian motion

Giang T. Nguyen; Oscar Peralta

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arXiv · 1908.11075

Rate of Strong Convergence to Markov-modulated Brownian motion

Giang T. Nguyen·Oscar Peralta

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Abstract

In Latouche and Nguyen (2015), the authors 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. We also 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 \cite{gorostiza1980rate}, which is $o(n^{-1/2}(\log n)^{5/2})$.

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BibTeX RIS

Giang T. Nguyen, Oscar Peralta. 2019-08-29. Rate of Strong Convergence to Markov-modulated Brownian motion. https://arxiv.org/abs/1908.11075

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