  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 $o(n^{-1/2}(\log
n)^{5/2})$.
