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Slides09-2010

# Slides09-2010 - Outline Martingales Stochastic Process...

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Unformatted text preview: Outline Martingales Stochastic Process Martingales and Random Walks: Lecture IX Charles B. Moss September 10, 2010 Charles B. Moss Martingales and Random Walks Outline Martingales Stochastic Process Martingales Stochastic Process Charles B. Moss Martingales and Random Walks Outline Martingales Stochastic Process Margingales I 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 ). I As with most of our models of risk, martingales have their basis in gambling concepts. I Let the random variable be the total winnings for the gambler after n games of chance ( X i : i = 1 , 2 , · · · ). I The question is then: what will the return be on the next game X n +1 . I Specifically, if the game is fair E [ X n + 1] (also, the expected winnings on all the previous games is also zero). I 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) Charles B. Moss Martingales and Random Walks Outline Martingales Stochastic Process I Definition 2.18 p-53 The sequence { ( X n , B...
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Slides09-2010 - Outline Martingales Stochastic Process...

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