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When the initial and transition probabilities of a finite Markov chain indiscrete time are not well known, we should perform a sensitivity analysis.This can be done by considering as basic uncertainty models the so-calledcredal sets that these probabilities are known or believed to belong to, and byallowing the probabilities to vary over such sets. This leads to the definitionof an imprecise Markov chain. We show that the time evolution of such a systemcan be studied very efficiently using so-called lower and upper expectations,which are equivalent mathematical representations of credal sets. We also studyhow the inferred credal set about the state at time n evolves as n goes toinfinity: under quite unrestrictive conditions, it converges to a uniquelyinvariant credal set, regardless of the credal set given for the initial state.This leads to a non-trivial generalisation of the classical Perron-FrobeniusTheorem to imprecise Markov chains.

Imprecise Markov chains and their limit behaviour