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:

**W**_{t} is a Brownian motion. Use Ito's Lemma to derive **dX = µ(W,t)dt + σ(W,t)dW**_{t} for the following function:

**X**_{t }= exp{ ∫_{0}^{t} θdW_{s} – 1/2 ∫_{0}^{t} θ^{2}ds }

Thanks so much in advance!!

Val

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).

1 Attachment(s)

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

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.