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The goal of this work is to formally abstract a Markov process evolving indiscrete time over a general state space as a finite-state Markov chain, withthe objective of precisely approximating its state probability distribution intime, which allows for its approximate, faster computation by that of theMarkov chain. The approach is based on formal abstractions and employs anarbitrary finite partition of the state space of the Markov process, and thecomputation of average transition probabilities between partition sets. Theabstraction technique is formal, in that it comes with guarantees on theintroduced approximation that depend on the diameters of the partitions: assuch, they can be tuned at will. Further in the case of Markov processes withunbounded state spaces, a procedure for precisely truncating the state spacewithin a compact set is provided, together with an error bound that depends onthe asymptotic properties of the transition kernel of the original process. Theoverall abstraction algorithm, which practically hinges on piecewise constantapproximations of the density functions of the Markov process, is extended tohigher-order function approximations: these can lead to improved error boundsand associated lower computational requirements. The approach is practicallytested to compute probabilistic invariance of the Markov process under study,and is compared to a known alternative approach from the literature.

Quantitative Approximation of the Probability Distribution of a Markov Process by Formal Abstractions