This capstone runs the full toolkit built across this topic — classification, stationary distributions, detailed balance, convergence, exit analysis — on three applications chosen for their reach beyond finance: a full end-to-end credit-migration calculation, the linear-algebra idea behind PageRank, and a first look at how a designed Markov chain can be used to sample from an arbitrary target distribution (the idea underlying Markov chain Monte Carlo, taken up properly in the discrete martingales topic's more advanced simulation methods and in portfolio credit risk's model-calibration workflows).
Learning objectives. By the end of this note you should be able to:
- carry a credit-migration matrix through classification, stationary-distribution, and exit-time analysis end-to-end;
- explain the linear-algebra idea behind PageRank as a stationary-distribution computation;
- state the Metropolis-Hastings acceptance rule and explain why it produces a chain with a prescribed stationary distribution.
In this lesson
- Credit migration, end to end
- PageRank as a stationary distribution
- An MCMC preview: the Metropolis-Hastings idea
The full lesson, with worked examples and exercise solutions, is for members.
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