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Optional stopping theorem - Wikipedia

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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. Since martingales can be used to model the wealth of a gambler participating in a fair game, the optional stopping theorem says that, on average, nothing can be gained by stopping play based on the information obtainable so far (i.e., without looking into the future). Certain conditions are necessary for this result to hold true. In particular, the theorem applies to doubling strategies. The optional stopping theorem is an important tool of mathematical finance in the context of the fundamental theorem of asset pricing. A discrete-time version of the theorem is given below, with 𝑁 0 denoting the set of natural integers, including zero. Let X = (Xt)t∈ 𝑁 0 be a discrete-time martingale and τ a stopping time with values

Optional stopping theorem - Wikipedia Jump to content From Wikipedia, the free encyclopedia Theorem in probability theory Not to be confused with Optimal stopping . In probability theory , the optional stopping theorem (or sometimes Doob's optional sampling theorem , for American probabilist Joseph Doob [ 1 ] ) says that, under certain conditions, the expected value of a martingale at a stopping time is equal to its initial expected value. [ 2 ] The concept can be understood through the following key principles: Since martingales can be used to model the wealth of a gambler participating in a

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