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Thread: Infinitesimal change in Wiener process proportional to the square root of time?

  1. #1
    Newbie mkmath's Avatar
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    Infinitesimal change in Wiener process proportional to the square root of time?

    I see the following statement made in a number of finance papers which indicates that the infinitesimal change in the Wiener process is proportional to the square root of (time), where the proportionality coefficient is normally distributed.

    Infinitesimal change in Wiener process proportional to the square root of time?-capture.png

    I cannot find a rigorous proof (or even a rudimentary proof) for this.

    Am I missing something? Is it the case that dW is somehow just chosen to be  N\sqrt{dt}.

    TIA
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  2. #2
    MHF Contributor
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    Re: Infinitesimal change in Wiener process proportional to the square root of time?

    Hey mkmath.

    There is a property of a Weiner process in terms Wt and Wt+n which you should use to get this result [and the n > 0].
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    Newbie mkmath's Avatar
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    Re: Infinitesimal change in Wiener process proportional to the square root of time?

    Thank you Chiro. I think my question is more fundamental than the Wiener process alone. For example, I can see how Var( \Delta W_t) = \Delta t from the following derivation:
    Infinitesimal change in Wiener process proportional to the square root of time?-capture.png

    So assuming that \Delta t \rightarrow dt for small time increments, then Var(  dW_t) = dt and \sigma(dW_t) \rightarrow \sqrt{dt}. At this point I am assuming the previous sentence is correct. If so, I don't understand what assumption/step is undertaken when to arrive at the statement dW = (some scaling factor )*\sigma(W_t).
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    Re: Infinitesimal change in Wiener process proportional to the square root of time?

    Try putting the dt factor within the variance term [since sqrt(dt)^2 = dt] and then bring it outside to show that you can have a*N(0,1) = N(0,a^2).
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