Hitting Times for Continuous-Time Imprecise-Markov Chains

Thomas Krak

Keywords: imprecise-Markov chains, hitting times, lower and upper expectations, robust inference

TL;DR: We show that the lower- and upper expected hitting times are the same for three types of imprecise-Markov chains, and that these satisfy particular non-linear systems of equations.

Abstract:

We study the problem of characterizing the expected hitting times for a robust generalization of continuous-time Markov chains. This generalization is based on the theory of imprecise probabilities, and the models with which we work essentially constitute sets of stochastic processes. Their inferences are tight lower- and upper bounds with respect to variation within these sets.

We consider three distinct types of these models, corresponding to different levels of generality and structural independence assumptions on the constituent processes.

Our main results are twofold; first, we demonstrate that the hitting times for all three types are equivalent. Moreover, we show that these inferences are described by a straightforward generalization of a well-known linear system of equations that characterizes expected hitting times for traditional time-homogeneous continuous-time Markov chains.