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Lecture 4: Continuous-time Markov Chains

Miranda Holmes-CerfonApplied Stochastic Analysis, Spring 2019 Lecture 4: Continuous-time Markov ChainsReadings Grimmett and Stirzaker (2001) , : Grimmett and Stirzaker (2001) (a survey of the issues one needs to address to make the discussionbelow rigorous) Norris (1997) Chapter 2,3 (rigorous, though readable; this is the classic text on Markov Chains , bothdiscrete and continuous ) Durrett (2012) Chapter 4 (straightforward introduction with lots of examples)Many random processes have a discrete state space , but can change their values at any instant of timerather than at fixed intervals: radioactive atoms decaying, the number of molecules in a chemical reaction,populations with birth/death/immigration/emigration, the number of emails in an inbox, etc.

4.1 Definition and Transition probabilities Definition. Let X =(X t) t 0 be a family of random variables taking values in a finite or countable state space S, which we can take to be a subset of the integers. X is a continuous-time Markov chain (ctMC) if it satisfies

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