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In probability theory, optional stopping theorem (or Doob's optional sampling theorem) says that, under certain conditions, the expected value of a martingale at a stopping time is equal to the expected value of its initial 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 the average nothing can be gained by stopping to play the game based on the information obtainable so far (i.e., by not looking into the future). Of course, certain conditions are necessary for this result to hold true, in particular doubling strategies have to be excluded. The optional stopping theorem is an important tool of mathematical finance in the context of the fundamental theorem of asset pricing. == Statement of theorem == A discrete-time version of the theorem is given below: Let be a discrete-time martingale and a stopping time with values in }, both with respect to a filtration . Assume that one of the following three conditions holds: :() The stopping time is almost surely bounded, i.e., there exists a constant such that a.s. :() The stopping time has finite expectation and the conditional expectations of the absolute value of the martingale increments are almost surely bounded, more precisely, and there exists a constant such that 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「optional stopping theorem」の詳細全文を読む スポンサード リンク
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