Path integration solutions for stochastic systems with Markovian jumps

A path integration method for solving Markovian jump stochastic dynamical systems is presented. The Markovian jump process and the state vector of the system are combined into a new augmented state vector. The randomness of the Markovian jump is modeled by a stochastic process, which is merged with the stochastic perturbations to form the stochastic source affecting the evolution of the augmented system. Then a probability mapping is constructed, mapping the probability space of the stochastic source to the short-time transition probability density function of the augmented system. By solving this probability mapping, the short-time transition probability density function of the path integration method is obtained, thus a path integration method is customized for Markovian jump stochastic systems. Finally, a hydraulic relief valve system is used as an application example to demonstrate the availability of the path integration algorithm. The results suggest that the pre-compression parameters and coefficient of restitution of the plug can induce stochastic P-bifurcation phenomena. When the above two parameters are small, the system becomes unstable with the parameters taken in this example. Monte Carlo simulations prove that the proposed method effectively captures the transient and stationary responses of Markovian jump systems.