  Hybrid stochastic differential equations are a useful tool to model
continuously varying stochastic systems which are modulated by a random
environment that may depend on the system state itself. In this paper, we
establish the pathwise convergence of the solutions to hybrid stochastic
differential equations via space-grid discretizations. While time-grid
discretizations are a classical approach for simulation purposes, our
space-grid discretization provides a link with multi-regime Markov modulated
Brownian motions, leading to computational tractability. We exploit our
convergence result to obtain efficient approximations to first passage
probabilities and expected occupation times of the solutions hybrid stochastic
differential equations, results which are the first of their kind for such a
robust framework. We finally illustrate the performance of the resulting
approximations in numerical examples.
