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Partial Differential Equations and Probability Density Functions

Please see the attached file for the fully formatted problems.
1) Suppose that S is a random variable that is defined on [0,∞) and whose probability density function is:

G(s) = ,

a and b being positive numbers. Show:

where
N(z) = .

2) We know that the solution of the final value problem

, 0 ≤ S, 0 ≤ t ≤ T,
0 ≤ S
is

Here
G(S',T;S,t) = ,
where
and .

Using this result and the result in 1), show that the price of a put option is given by:

where
,
.

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Solution Summary

Partial differential equations and PDFs are investigated.

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