The Limit Behaviour of Imprecise Continuous-Time Markov Chains

Abstract

We study the limit behaviour of a nonlinear differential equation whose solution is a superadditive generalisation of a stochastic matrix, prove convergence, and provide necessary and sufficient conditions for ergodicity. 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 nonlinear 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 convergence result shows that for a fixed initial state, an imprecise continuous-time Markov chain always converges to a limiting distribution, and our ergodicity result provides a necessary and sufficient condition for this limiting distribution to be independent of the initial state.