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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

ξ = ,

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

d ξ = a(ξ)dt + b(ξ)dX.

Find the concrete expressions for a(ξ) and b(ξ) and show that a(ξ) and b(ξ) 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(ξ) and b(ξ). 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. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.