The Limit Behaviour of Imprecise Continuous-Time Markov Chains

Probability (math.PR) FOS: Mathematics 0202 electrical engineering, electronic engineering, information engineering 02 engineering and technology Mathematics - Probability
DOI: 10.1007/s00332-016-9328-3 Publication Date: 2016-08-29T11:40:52Z
ABSTRACT
We study the limit behaviour of a generally non-linear ordinary differential equation whose solution is a superadditive generalisation of a stochastic matrix, and provide necessary and sufficient conditions for this solution to be ergodic, in the sense that it converges to an operator that, essentially, maps functions to constants. In the linear case, the solution of our differential equation is equal to the matrix exponential of an intensity matrix and can then be interpreted as the transition operator of a homogeneous continuous-time Markov chain. Similarly, in the generalised non-linear case that we consider, the solution can be interpreted as the lower transition operator of a specific set of non-homogeneous continuous-time Markov chains, called an imprecise continuous-time Markov chain. In this context, our main result provides a necessary and sufficient condition for such an imprecise continuous-time Markov chain to converge to a unique limiting distribution.
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