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Math Help - Stochastic differential equation

  1. #1
    Super Member Deadstar's Avatar
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    Stochastic differential equation

    Consider the stochastic differential equation

    d S_t = \mu S_t dt + \sigma^2 S_t dW_t for all t \in [0,T]

    where \{ W_t \}t_0 is a standard Brownian motion, \mu, \sigma and S_0 are positive constants. Derive the unique solution to the above SDE.

    Apparently this was derived in class but I'm missing that lecture so I guess I wasn't there or have lost it...

    The actual solution is...

    S_t = S_0 e^{\left (\mu - \frac{\sigma^2}{2} \right) t + \sigma W_t}

    Anyone got any ideas on how to derive that?

    First step is...

    S_t = s + \mu \int_0^T S_u du + \sigma \int_0^T S_u dW_u

    I seem to think it involves Ito process/lemmas...

    Maybe I should do something like add in...

    \frac{\sigma^2}{2} - \frac{\sigma^2}{2}
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  2. #2
    Super Member Deadstar's Avatar
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    Ok I found another answer elsewhere but I still have questions...

    http://mplab.ucsd.edu/tutorials/sde.pdf (end of pg 13-14)

    Here is the solution I found except I replace X_t with S_t and B_t with W_t

    Using Ito's rule on \log(S_t) we get (Why \log(S_t)???).
    <br />
d \log S_t = \frac{1}{S_t}dS_t + \frac{1}{2} \left ( -\frac{1}{S_t^2} \right )^2 (dS_t)^2

    = \frac{dS_t}{S_t} - \frac{1}{2} \sigma^2 dt


    = \left ( \mu S_t - \frac{1}{2} \sigma^2 \right )dt + \alpha dW_t

    Thus, And thus solution...
    \log S_t = \log S_0 + \left ( \mu - \frac{1}{2}\sigma^2 \right )t + \sigma W_t

    And thus solution...

    I honestly can't justify any of those steps.
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  3. #3
    Super Member Deadstar's Avatar
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    Ok perhaps I can justify the first step.

    Ito lemma (process?)

    dF(x_t) = F'(X_t)g_tdt + F'(X_t)h_tdW_t + \frac{1}{2}F''(X_t)d_t^2 dt as written in my notes.

    So!

    d \log S_t = \mu \frac{1}{S_t} dS_t + \sigma \frac{1}{S_t} dW_t + \frac{1}{2}\left (-\frac{1}{S_t^2} \right )^2 (dS_t)^2

    But why does the middle term disappear (I guess it is = to 0 somehow that I'm not seeing)
    And what happens to the \mu and \sigma terms? Why were they left out at this step in the solution I found.
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