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Partial Differential Equations : Wiener Process and Ito's Lemma

Please help with the following problems. See the following posting for the complete equations

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 ξ with respect to r, I do not know how to treat the absolute value sign. Thanks.


Solution Summary

Wiener Processes and Ito's Lemma are investigated in the following posting. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.