In this paper, we focus on the mean exit time and the scale function for the geometric Brownian motion with Markovian switching, in which the drift coefficients and the diffusion coefficients are associated with regime changes. The explicit expressions of mean exit time and scale function are obtained by solving the corresponding Poisson problem. Furthermore, we estimate parameters for the geometric Brownian motion with Markovian switching by composite likelihood and explore some properties of the estimates. Computer simulations are performed to illustrate our proposed algorithm, showing high accuracy of the estimators.