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Math Help - Derivation for Dolean's Exponential using Ito's Lemma

  1. #1
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    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 = (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
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  2. #2
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    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).
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  3. #3
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    Re: Derivation for Dolean's Exponential using Ito's Lemma

    Quote Originally Posted by chiro View Post
    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?

    Derivation for Dolean's Exponential using Ito's Lemma-ito-s-lemma.jpg
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  4. #4
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    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.
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