Quantifying and Managing Uncertainty in Piecewise-Deterministic Markov Processes

Abstract

In piecewise-deterministic Markov processes (PDMPs) the state of a finite-dimensional system evolves continuously, 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) of the cumulative cost 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 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 for several one-dimensional and two-dimensional first-exit time problems. In the appendix, we also apply this method to a model of fish harvesting in an environment with random switches in carrying capacity.