Under suitable regularity conditions, the expected value of a martingale evaluated at a random stopping time equals its expected initial value. It formalizes the intuitive 'no free lunch' principle behind fair gambling strategies and is central to modern probability and mathematical finance.
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StatementUnder suitable regularity conditions on the stopping time and the martingale, such as bounded stopping times or uniform integrability, the expected value of the martingale evaluated at the stopping time equals its initial expected value. 1 Classification
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Source Optional Stopping Theorem (Wikipedia)
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1. Optional Stopping Theorem (Wikipedia)
Wikimedia FoundationLead section
In probability theory, the optional stopping theorem (or sometimes Doob's optional sampling theorem, for American probabilist Joseph Doob) says that, under certain conditions, the expected value of a martingale at a stopping time is equal to its initial expected value.
Proved By: Joseph L. Doob, Lead paragraph
In probability theory, the optional stopping theorem (or sometimes Doob's optional sampling theorem, for American probabilist Joseph Doob) says that, under certain conditions, the expected value
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