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Contents of PMS, Vol. 39, Fasc. 2,
pages 259 - 277
DOI: 10.19195/0208-4147.39.2.2
 

EMBEDDED MARKOV CHAIN APPROXIMATIONS IN SKOROKHOD TOPOLOGIES

Björn Böttcher

Abstract: We prove a J 1 -tightness condition for embedded Markov chains and discuss four Skorokhod topologies in a unified manner.

To approximate a continuous time stochastic process by discrete time Markov chains, one has several options to embed the Markov chains into continuous time processes. On the one hand, there is a Markov embedding which uses exponential waiting times. On the other hand, each Skorokhod topology naturally suggests a certain embedding. These are the step function embedding for J 1 , the linear interpolation embedding for M 1 , the multistep embedding for J 2 and a more general embedding for M 2 . We show that the convergence of the step function embedding in J 1 implies the convergence of the other embeddings in the corresponding topologies. For the converse statement, a J 1 -tightness condition for embedded time-homogeneous Markov chains is given.

Additionally, it is shown that J 1 convergence is equivalent to the joint convergence in M 1 and J 2 .

2000 AMS Mathematics Subject Classification: Primary: 60B10; Secondary: 60J75, 60J05.

Keywords and phrases: Markov chain embedding, tightness, Skorokhod space, Skorokhod topologies, jump processes, Markov chain approximation.

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