Actuarial Outpost Ito Lemma question
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 Investment / Financial Markets Old Exam MFE Forum

#1
07-07-2014, 10:42 AM
 Hawthornen SOA AAA Join Date: Apr 2014 College: Purdue University Alumni Posts: 4
Ito Lemma question

So I've been struggling with a somewhat fundamental part of Ito's lemma and maybe someone could help explain what's going on.

In one of my sample problems (changed slightly)

dS(t)/S(t) = .3dt + .4dZ(t)
G(t) = S(t)^t

And it wants me to basically find a stochastic differential equations satisfied by G(t)

When I attempt to work out the partial derivatives I feel like the variables I differentiate with regard to are opposite to that in the solution.

For example: the partial derivative with regard to S(t) is listed as t*S(t)^(t-1)
and the partial with regard to t is ln(S)*S^t

To me this always seems backwards. Maybe I'm missing something or just looking at it wrong but my first partial derivatives are correct but for the wrong variables (so my second partial is always incorrect). Maybe someone could explain where I'm going wrong.
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Lazy Exam Checklist
P -> FM -> MLC -> MFE -> C
#2
07-07-2014, 10:57 AM
 kernel Member SOA Join Date: Mar 2010 Posts: 843

Ask yourself what the derivative of the following would be from a calculus one class:

1. y = x^5

2. y = 5^x
#3
07-07-2014, 12:39 PM
 mono/poly Member SOA Join Date: Jul 2013 Posts: 411

Partial derivative with respect to S --> treat S as a variable, t as a numerical constant
Partial derivative with respect to t --> treat t as a variable, S as a numerical constant
#4
07-07-2014, 01:27 PM
 Hawthornen SOA AAA Join Date: Apr 2014 College: Purdue University Alumni Posts: 4

Holy crud thanks guys. I've done plenty of partials. I must be just tired or something, I don't know why I was getting that so backwards.

Appreciate the help.
__________________
Lazy Exam Checklist
P -> FM -> MLC -> MFE -> C

Last edited by Hawthornen; 07-08-2014 at 09:12 AM..