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

TL;DR

This paper develops a space-grid discretization method for hybrid stochastic differential equations, enabling efficient computation of first passage probabilities and occupation times, with proven convergence and practical numerical demonstrations.

Contribution

It introduces a novel space-grid approximation approach for hybrid SDEs, linking to Markov modulated processes and providing the first computational methods for first passage properties.

Findings

Proves pathwise convergence of space-grid discretizations.

Provides efficient algorithms for first passage probabilities.

Demonstrates numerical accuracy through examples.

Abstract

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.