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# Partial Differential Equations : Wiener Process and Ito's Lemma

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Consider a random variable r satisfying the stochastic differential equation:

Where are positive constants and dX is a Wiener process. Let

(see attached file)

which transforms the domain for r into (-1,1) for Xi Suppose the stochastic equation for the new random variable Xi is in the form:

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

Find the concrete expressions for a(Xi) and b(Xi) and show that a(Xi) and b(Xi) fulfill the conditions:

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

I am confused because I don't know if you can use Ito's lemma to find the concrete expressions for a(Xi) and b(Xi). Also when taking the partial derivative of &#958; with respect to r, I do not know how to treat the absolute value sign. Thanks.