Lecture09-2010 - Martingales and Random Walks: Lecture IX...

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Unformatted text preview: Martingales and Random Walks: Lecture IX Charles B. Moss September 10, 2010 I. Martingales A. Suppose that we have a sequence of random variables X t : t = 1 , 2 , (or X 1 , X 2 , X 3 ) defined on a measure space ( C , B , P ). 1. As with most of our models of risk, martingales have their basis in gambling concepts. 2. Let the random variable be the total winnings for the gambler after n games of chance ( X i : i = 1 , 2 , ). 3. The question is then: what will the return be on the next game X n +1 . 4. Specifically, if the game is fair E [ X n + 1] (also, the expected winnings on all the previous games is also zero). 5. Thus, if the game is fair this result is independent of past winnings. Mathematically E [ X n +1 | X 1 , X n ] = X n (1) B. Definition 2.18 p-53 The sequence { ( X n , B n ) , n = 1 , 2 , } is a Martingale if each n satisfies 1. C n C n +1 or the Borel sets are nested. Thus, the set con- taining the current draw are contained in the set containing...
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Lecture09-2010 - Martingales and Random Walks: Lecture IX...

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