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
,
.

Attached file(s):

pde 1-2.doc  View File