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Math Help - maximize a sequence of functions

  1. #1
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    maximize a sequence of functions

    Hallo,

    Someone already helped me to show that the following convergence is uniformly for all  a \in [0,1]:
    \lim\limits_{h \to 0}\left(1+a(\mu-r)h+\frac{1}{2}a^2 \gamma(1-\gamma)\sigma^2h+O(h^\frac{3}{2})\right)^\frac{1}{  h}=e^{a(\mu-r)+ \frac{1}{2}a^2 \gamma(1-\gamma)\sigma^2}.

    whereas
    O(h^\frac{3}{2}) is the remainder term of a taylor expansion where all terms with factor h^\frac{3}{2} and higher exponents are collected. This term can be bounded by a constant that can be chosen independent of a
    \mu \in \mathbb{R} is a constants
    \sigma,r >0 are constants
    0<\gamma <1 is a constant
    f_h(a):=\left(1+a(\mu-r)h+\frac{1}{2}a^2  \gamma(1-\gamma)\sigma^2h+Ch^\frac{3}{2})\right)^\frac{1}{h  }
    f(a):=e^{a(\mu-r)+  \frac{1}{2}a^2 \gamma(1-\gamma)\sigma^2}


    does the maximum point of the sequence of functions (f_h(a))_{h \in (0,1)} (I call it x_h) with a \in [0,1] converge to the maximum point of f for (I call it x) h\to 0?


    Does anyone know if the following holds:
    \lim\limits_{h \to 0}x_h = x?



    I only know a theorem which says that this is true for a sequence of concave functions. But my functions are neither concave nor convex for all a \in [0,1].

    Can anybody help me?
    Thanks!
    Last edited by Juju; March 18th 2011 at 01:29 PM.
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  2. #2
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    I know that (f_h(a))^h is concave in a. So there's a unique maximum point of (f_h(a))^h on [0,1].
    Since the function x^\frac{1}{h}, x\geq0 is strictly increasing, (f_h(a))^h and f_h(a) have the same maximum point.

    Can I now conclude that f_h(a) has also a unique maximum point?

    Also \log_e(f(a)) is concave in a and e^x is strictly increasing.
    Then the maximum point of f(a) on [0,1] is unique. Am I right?

    If the maximum points of f_h and f are unique I think it follows from the uniform convergence that it holds
    x_h \to x.
    Am I right?

    Sorry, I made some mistakes in the definitions. There should be a (\gamma-1) instead of the (1-\gamma). So, the corrected version:
    f_h(a):=\left(1+a(\mu-r)h+\frac{1}{2}a^2  \gamma(\gamma-1)\sigma^2h+O(h^\frac{3}{2})\right)^\frac{1}{h}
    f(a):=e^{a(\mu-r)+  \frac{1}{2}a^2 \gamma(\gamma-1)\sigma^2}
    Last edited by Juju; March 19th 2011 at 12:38 PM.
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