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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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3.2 Problems and Solutions

3.2.1 Itō Calculus

  1. 1. Itō Integral. Let c03-math-172 be a probability space and let c03-math-173 be a standard Wiener process. Let the Itō integral of c03-math-174 be defined as the following limit
    equation

    where c03-math-175, c03-math-176 for c03-math-177.

    Show that the quadratic variation of c03-math-178 is

    equation

    and hence

    equation

    Finally, show that the Itō integral is a martingale.

    Solution

    For the first part of the solution, see Problem 2.2.6.1 (page 89).

    Given c03-math-179 and ...

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