Uncertainty propagation of stochastic hybrid systems: a case study for types of jump

Stochastic hybrid systems are dynamic systems that undergo both random continuous-time flows and random discrete jumps. Depending on how randomness is introduced into the continuous dynamics, discrete transitions, or both, stochastic hybrid systems exhibit distinct characteristics. This paper investigates the role of uncertainties in the interplay between continuous flows and discrete jumps by studying probability density propagation. Specifically, we formulate stochastic Koopman/Frobenius-Perron operators for three types of one-dimensional stochastic hybrid systems to uncover their unique dynamic characteristics and verify them using Monte Carlo simulations.