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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 Problems and Solutions

2.2.1 Basic Properties

  1. 1. Let c02-math-095 be a probability space. We consider a symmetric random walk such that the c02-math-096-th step is defined as
    equation

    where c02-math-097, c02-math-098. By setting c02-math-099, we let

    equation

    where c02-math-100.

    Show that the symmetric random walk has independent increments such that the random variables

    equation

    are independent.

    Finally, show that c02-math-101 and c02-math-102.

    Solution

    By definition

    and for , ,

    Because and since is independent ...

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