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Hybrid stochastic differential equations are a useful tool to modelcontinuously varying stochastic systems which are modulated by a randomenvironment that may depend on the system state itself. In this paper, weestablish the pathwise convergence of the solutions to hybrid stochasticdifferential equations via space-grid discretizations. While time-griddiscretizations are a classical approach for simulation purposes, ourspace-grid discretization provides a link with multi-regime Markov modulatedBrownian motions, leading to computational tractability. We exploit ourconvergence result to obtain efficient approximations to first passageprobabilities and expected occupation times of the solutions hybrid stochasticdifferential equations, results which are the first of their kind for such arobust framework. We finally illustrate the performance of the resultingapproximations in numerical examples.

Space-grid approximations of hybrid stochastic differential equations and first passage properties