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Problems and Solutions in Mathematical Finance: Stochastic Calculus, Volume I by Eric Chin, Sverrir Olafsson, Dian Nel

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2.2.4 First Passage Time

  1. 1. Doob's Maximal Inequality. Let c02-math-554 be a probability space and let c02-math-555 be a continuous non-negative submartingale with respect to the filtration c02-math-556, c02-math-557. Given c02-math-558 and c02-math-559, show that
    equation

    By writing

    equation

    show that

    equation

    Deduce that if c02-math-560 is a continuous non-negative supermartingale with respect to the filtration c02-math-561, then

    Solution

    For we let so that . Because is a non-negative submartingale we have ...

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