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    Partial Differential Equations : Wiener Process

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    Please see the attached file for the fully formatted problems.
    1) Suppose: dS = a(S,t)dt + b(S,t)dX,

    where dX is a Wiener process. Let f be a function of S and t.
    Show that:
    (see the attached file for equations)
    2) Suppose that S satisfies

    (see the attached file for equations)

    where u >=0, signa> 0, and dX is a Wiener process. Let

    Xi = S/(S + Pm)

    where Pm is a positive constant and the range of &#958; is [0,1), if 0 &#8804; S < &#8734;. The stochastic differential equation for &#958; is in the form:

    d(Xi)= a(Xi)dt + b(Xi)dX.

    Find the concrete expressions for a(Xi) and b(Xi) by Ito's lemma and show:

    {a(0) = 0, and {a(1) = 0,
    {b(0) = 0, and {b(1) = 0.

    Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach by Duffy. See attached file for full problem description.

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    Solution Summary

    Partial Differential Equations and Wiener Processes are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.