I suppose most of the college-aged folks are already past their finals by now, but what would THF be without a Math Help Thread?!

Hit it up, boyz.

I never took Stochastic Calculus, so apologies for that...

I did however find this: http://en.wikipedia.org/wiki/Geometric_Brownian_motion#Solving_the_SDE

That and the "Properties" section both show solutions to what the first part of question 2. You just need to do some algebra to get it into the form shown in the exam by taking exponentials of both sides.

What's your question here?

Are you just asking why h'(x) is what it is? That's a fairly common derivative...

If h(x) = a ^x, h'(x) = a^x * ln(a)

Edit: That was a terrible identity; I'll just go this the long way.

if you set y = a^x, by taking ln() of both sides, you get ln(y) = x * ln(a)

Differentiating, we have dy/dx * 1/y = ln(a). Multiple both sides by y, and replace y(or h(x) as it is in the problem) with a^x, and you're left with h'(x) = ln(a)*a^x

Oh and probably also worth nothing that 1/(a^x) = a^(-x)