Space-grid approximations of hybrid stochastic differential equations and their ruin probabilities

Abstract: Hybrid stochastic differential equations are a useful tool for modeling continuously varying stochastic systems which are modulated by a random environment that may depend on the system state itself. In this talk, we establish the pathwise convergence of the solutions to hybrid stochastic differential equations through 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 for the ruin times and expected occupation times of the solutions of hybrid stochastic differential equations used to model risk reserve processes, results that are the first of their kind for such a robust framework.

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