Applied Probability (MT 2011)

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By: Neil Laws

Summary of the course

The theoretical content of this course is concerned with two generalisations of the Poisson process.

- The construction of the Poisson process in terms of exponential holding times and the resulting Markov property can be considerably generalised. This is done es- sentially by allowing different parameters for the holding times in different states and allowing jumps, which instead of always being +1, are random and depend on the current state. This gives the class of continuous-time Markov chains. We will spend roughly the first half of the course studying continuous-time Markov chains. Our main reference will be Markov Chains by Norris.

- The Poisson process is the prototype of a counting process. Many quantities can be explicitly calculated for it. However, in applications, exponential inter-arrival times may not be a appropriate, for example when modelling the arrival of insurance claims. If we relax the assumption of exponentiality of the inter-arrival times (but keep their independence and identical distribution) we obtain the class of counting processes called renewal processes. Since exact calculations are often impossible or not helpful in this context, the most important results of renewal theory are limiting results. Our main reference will be Chapter 10 of Probability and Random Processes by Grimmett and Stirzaker.

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