# Derivation for Dolean's Exponential using Ito's Lemma

• Feb 5th 2013, 05:00 PM
valiant
Derivation for Dolean's Exponential using Ito's Lemma
Hi all,

Been trying to make sense of this for a couple days now, but my background in calculus is a little light.... Taking a financial engineering class and trying to work a solution for the following:

Wt is a Brownian motion. Use Ito's Lemma to derive dX = µ(W,t)dt + σ(W,t)dWt for the following function:
Xt = exp{ ∫0t θdWs – 1/2 ∫0t θ2ds }

Thanks so much in advance!!
Val
• Feb 5th 2013, 10:51 PM
chiro
Re: Derivation for Dolean's Exponential using Ito's Lemma
Hey valiant.

Are you given the rules for the Ito lemma to relate the integrals of the random and deterministic parts? (Once you have this then it becomes a match and fudge game).
• Feb 6th 2013, 06:37 AM
valiant
Re: Derivation for Dolean's Exponential using Ito's Lemma
Quote:

Originally Posted by chiro
Hey valiant.

Are you given the rules for the Ito lemma to relate the integrals of the random and deterministic parts? (Once you have this then it becomes a match and fudge game).

Hi Chiro,

Not sure I completely understand the "rules" you're talking about. Do you mean the derivation of Ito's Lemma we're using? If so, the attached picture shows the form our professor gave us in class. Was that what you were looking for?

Attachment 26880
• Feb 6th 2013, 12:35 PM
chiro
Re: Derivation for Dolean's Exponential using Ito's Lemma
No there should be a general version in terms of expressing X(t) as a function of two integrals: one with a Wiener component and the other with a deterministic component: both for general functions that modulate the dWt and the dt components of the the original dXt (which you have written there).

I can't remember it off hand since it is pretty complex.