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Using Itos lemma to find an SDE

  1. Feb 11, 2013 #1
    1. The problem statement, all variables and given/known data
    Using Ito's lemma to find the SDE satisfied by ##U## given that ##U = ln(Y)## and ##Y## satisfies:
    [tex]
    dY = \frac{1}{2Y}dt + dW
    [/tex]

    [tex]
    Y(0) = Y_0
    [/tex]

    2. Relevant equations

    Ito's lemma.

    3. The attempt at a solution

    If ##U \equiv U(Y,t)## and ##dY = a(Y,t)dt + b(Y,t)dW##

    Then ##dU = \left(\frac{\partial U}{\partial t} + a(Y,t)\frac{\partial U}{\partial Y} + \frac{1}{2}b(Y,t)^2\frac{\partial^2U}{\partial Y^2}\right)dt + b(Y,t)\frac{\partial U}{\partial Y}dW##

    Here ##U = ln(Y)## and ##dY = \frac{1}{2Y}dt + dW##

    ##\frac{\partial U}{\partial t}=0##, ##\frac{\partial U}{\partial Y} = \frac{1}{Y}##, ##\frac{\partial^2 U}{\partial Y^2} = \frac{-1}{Y^2}##

    Therefore:

    [tex]
    dU = \left(0 + \frac{1}{2Y^2} - \frac{1}{2Y^2} \right)dt + \frac{1}{Y}dW
    [/tex]

    [tex]
    dU = \frac{1}{Y}dW
    [/tex]

    I *think* i've arrived at the right answer but given that I haven't done any kind of mathematics for nearly two decades, I wonder if somebody would be so kind as to critique my approach and let me know if the answer is right?

    Any help, much appreciated.
     
  2. jcsd
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