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Math Help - upper bound for a random variable

  1. #1
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    upper bound for a random variable

    Hallo,
    I've to find a constant C \in \mathbb{R}_+ such that |Y_n|\leq C for a random variable Y_n which is defined by

    Y_n:=\frac{2 \cdot (\pi e^{\theta (\mu_n h+ \sigma (W_{nh}-W_{(n-1)h})})^3}{(1+\pi( e^{\theta (\mu_n h+ \sigma (W_{nh}-W_{(n-1)h}))}-1))^3}-\frac{ (\pi e^{\theta (\mu_n h+ \sigma  (W_{nh}-W_{(n-1)h})})^2}{(1+\pi( e^{\theta (\mu_n h+ \sigma  (W_{nh}-W_{(n-1)h}))}-1))^2}

    whereas
    W is a standard Brownian motion
    \mu is a normal distributed random variable
    \pi and \theta are random variables with values in the interval [0,1]
    0<h<1 is a parameter
    \sigma>0 is a constant

    Can anybody help me finding C such that |Y_n|\leq C?
    Thanks a lot!!
    Last edited by Juju; March 9th 2011 at 04:22 AM.
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  2. #2
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    If it helps, the vast majority of the details you provided are completely irrelevant. If I did it correctly, it is very easy to show |Y_n| \le 3. Just call the exponential term X_n.
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  3. #3
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    Thank you!
    I've done the calculation again and now it is clear. For the first term it holds
    Y_n:=\frac{2 \cdot (\pi e^{\theta (\mu_n h+ \sigma (W_{nh}-W_{(n-1)h})})^3}{(1+\pi( e^{\theta (\mu_n h+ \sigma (W_{nh}-W_{(n-1)h}))}-1))^3}\leq \frac{2 \cdot (\pi e^{\theta (\mu_n h+ \sigma (W_{nh}-W_{(n-1)h})})^3}{(\pi e^{\theta (\mu_n h+ \sigma (W_{nh}-W_{(n-1)h}))})^3}=2.
    For the second term I can do the same.
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