In piecewise-deterministic Markov processes (PDMPs) the state of a finite-dimensional system evolves dynamically, but the evolutive equation may change as a result of discrete switches that occur at random times. The total cost of the process is found by integrating a running cost along the corresponding piecewise-deterministic trajectory. We show how to compute the Cumulative Distribution Function (CDF) of this total cost. In the case that the PDMP is controlled in real time, we further show how to select a control to optimize that CDF. In both 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.

Joint work with Antonio Farah, April Nellis, Jacob Van Hook, and Alex Vladimirsky.
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