Skorokhod's Embedding Theorem
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High Quality Content by WIKIPEDIA articles! Let X be a real-valued random variable with expected value 0 and finite variance; let W denote a canonical real-valued Wiener process. Then there is a stopping time (with respect to the natural filtration of W), , such that W has the same distribution as X, mathbb{E}[tau] = mathbb{E}[X^{2}] and mathbb{E}[tau^{2}] leq 4 mathbb{E}[X^{4}]. (Naturally, the above inequality is trivial unless X has finite fourth moment.)

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High Quality Content by WIKIPEDIA articles! Let X be a real-valued random variable with expected value 0 and finite variance; let W denote a canonical real-valued Wiener process. Then there is a stopping time (with respect to the natural filtration of W), , such that W has the same distribution as X, mathbb{E}[tau] = mathbb{E}[X^{2}] and mathbb{E}[tau^{2}] leq 4 mathbb{E}[X^{4}]. (Naturally, the above inequality is trivial unless X has finite fourth moment.)