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

Consider a random variable r satisfying the stochastic differential equation:

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

&#958; = ,

which transforms the domain for r into (-1,1) for &#958;. Suppose the stochastic equation for the new random variable &#958; is in the form:

d &#958; = a(&#958;)dt + b(&#958;)dX.

Find the concrete expressions for a(&#958;) and b(&#958;) and show that a(&#958;) and b(&#958;) 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(&#958;) and b(&#958;). Also when taking the partial derivative of &#958; 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. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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