Some characterizations for Brownian motion with Markov switching

In this paper, we focus on some properties and the maximum distribution estimates for one-dimensional Brownian motion with Markov switching. The explicit expressions for density functions, the mean exit time and Laplace transform of the exit time are obtained by solving the corresponding Poisson problem. The results of this paper disclose the impact on mean exit time and the Laplace transform of the exit time as σ 1 tends to σ 2 . Furthermore, an appropriate upper bound and an appropriate lower bound on the probabilities are given for Brownian motion with Markov switching.