Like Markov processes, a weirdly useful class of stochastic processes. Often you can find a martingale within some stochastic process, or construct a martingale from a stochastic process and prove something nifty thereby; This idea connects and solves a bunch of tricky problems at once.
TODO: examples, maybe a CLT and something else wacky like the life table estimators of (Aalen 1978).
I am indebted to Saif Syed for setting my head straight about the utility of martingales, and Kevin Ross who, in part of Amir Dembo’s course materials, was the one whose explanation of the orthogonality interpretation of martingales finally communicated the neatness of this idea to me.
TBC.
Local martingales
The classical gambling strategy (double-down until you win) is in fact a local martingale.
References
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