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.
Facts
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 Connections
Sources
1. Optional Stopping Theorem (Wikipedia)
Wikimedia Foundationlead section, first paragraphQuote, lead section, first paragraph
In probability theory, the optional stopping theorem says that, under certain conditions, the expected value of a martingale at a stopping time is equal to its initial expected value.
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