Definition
A Doob martingale (named after J. L. Doob) is a generic construction that is always a martingale. Specifically, consider any set of random variables
taking values in a set for which we are interested in the function and define:
where the above expectation is itself a random quantity since the expectation is only taken over
and
are treated as random variables. It is possible to show that is always a martingale regardless of the properties of .
The sequence is the Doob martigale for f.
Read more about this topic: Doob Martingale
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