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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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  1. 14. Generalised Itō Integral. Let c03-math-350 be a probability space and let c03-math-351 be a standard Wiener process. Given that c03-math-352 is a simple process, show
    equation

    and

    equation

    Solution

    For the first result, using Taylor's theorem on c03-math-353 and subsequently applying Itō's formula we have

    equation

    Taking integrals from c03-math-354 to c03-math-355,

    equation

    and rearranging the terms, finally

    equation

    since .

    As for the second result, from Taylor's theorem and Itō's formula

    Taking integrals ...

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