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Partial Differential Equations : Wiener Process

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.


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.