A piecewise deterministic discrete-time model with simultaneous random jumps

In this paper, we propose a new stochastic discrete-time model, which is inspired by the established model of a continuous-time piecewise deterministic Markov process. The discrete-time dynamics consist of deterministic motion and random jumps. The deterministic motion is given by the direction of a drift vector field. The random jumps are triggered by a finite number of independent count processes and the corresponding state transitions are given by jump vector fields. One may think of the discrete-time system as an input-affine with drift, where the count processes act as inputs in the directions of the jump vector fields. In each time step, we allow an activation of multiple jump vector fields, which is different from the corresponding continuous-time model in which, almost surely, only one jump occurs at each time instant. The proposed discrete-time model also includes a user-prescribed step size, which allows approximations of continuous-time processes. In the limit of vanishing step size, the sample paths of the discrete-time model converge almost surely to the sample paths of a continuous-time piecewise deterministic Markov process. We also characterize mean-square exponential stability of the model under the assumption of linear drift and jump vector fields. The findings are applied to a linear discrete-time state-feedback system with random state measurements.