Quantifying and managing uncertainty in piecewise-deterministic Markov processes

In piecewise-deterministic Markov processes (PDMP) the state of a finite-dimensional system evolves dynamically, but the evolutive equation may change randomly as a result of discrete switches. A running cost is integrated along the corresponding piecewise-deterministic trajectory up to the termination to produce the cumulative cost of the process. We address three natural questions related to uncertainty in cumulative cost of PDMP models: (1) how to compute the Cumulative Distribution Function (CDF) when the switching rates are fully known; (2) how to accurately bound the CDF when the switching rates are uncertain; and (3) assuming the PDMP is controlled, how to select a control to optimize that CDF. In all three cases, our approach requires posing a (weakly-coupled) system of suitable hyperbolic partial differential equations, which are then solved numerically on an augmented state space. We illustrate our method using simple examples of trajectory planning under uncertainty.