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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. 24. Let c05-math-776 be a probability space and let c05-math-777 be a Poisson process with intensity c05-math-778 and c05-math-779 be a standard Wiener process relative to the same filtration c05-math-780, c05-math-781. By considering the process
    equation

    where c05-math-782, use Itō's formula to find an SDE for c05-math-783.

    By setting c05-math-784 show that solution to the SDE can be expressed as

    equation

    Solve the differential equation to find and deduce that .

    Solution

    By expanding using Taylor's theorem and taking ...

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