The random walk had one restrictive feature: its step distribution never changed, no matter where it currently stood. A credit-rating migration model needs more than that — a bond rated AA moves to AAA or A with different probabilities than a bond rated B moves to BB or CCC. A Markov chain is exactly this generalisation: the next state's distribution is allowed to depend on the current state, as long as it depends on nothing earlier than the current state. This single relaxation — "the rule for the next step can depend on where you are now" — is what turns the random walk into the general-purpose modelling object behind credit migration, weather forecasting, market-regime models, queueing systems, and, eventually, the theory of continuous-time processes that closes this unit.
Learning objectives. By the end of this note you should be able to:
- state the Markov property precisely and verify it (or its failure) for a proposed model;
- read off a transition matrix from a verbal description of a chain, and check that it is a valid transition matrix;
- represent a chain via its transition diagram;
- recognise the two defining properties every transition matrix must satisfy.
In this lesson
- The Markov property
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