Results 1 to 3 of 3

Math Help - what is quadratic vatiation

  1. #1
    Member
    Joined
    Nov 2006
    Posts
    136

    what is quadratic vatiation

    hi all, Ive searched online for ages but even though i can find formulas and definitions i dont quite understand what quadratic variation is.

    The reason i want to know this is, i'm looking to understand the importance of the fact that the quadratic variation of a Brownian motion process is t

    i.e quadratic variation of

    B[0,t] = t

    does anyone know how to explain this?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by chogo View Post
    hi all, Ive searched online for ages but even though i can find formulas and definitions i dont quite understand what quadratic variation is.
    I know absolutely nothing about Brownian motion ( I wish I did) but I looked at Wikipedia, it made some sense how they explain it.

    Heir.

    It looks like a variation of the Riemann integral to me.

    Let me give an example.
    Say we want to find the quadradic variation of f(t)=t on the interval [0,1]

    So we partition the region (like in the Riemann integral), I am going to be using left-endpoints because that is what they (Wikipedia) use.

    Thus, divide the region into "n" parts. Each part has 1/n width.

    The initial value is
    x_0=0
    x_1=0+1/n
    x_2=0+1/n+1/n
    x_3=0+1/n+1/n+1/n
    ...
    You get the idea.
    Thus,
    x_k = k/n
    Where 0<=k<=n-1
    (Because of left-endpoints).

    Now,
    [f(x_1)-f(x_0)]^2=(1/n-0/n)^2=1/n^2
    [f(x_2)-f(x_1)]^2=(2/n-1/n)^2=1/n^2
    [f(x_3)-f(x_2)]^2=(3/n-2/n)^2=1/n^2
    ...
    [f(x_{n-1}-f_{n-2}]^2=[(n-1)/n-(n-2)/n]^2=1/n^2

    Now, we sum them up,
    <f>_1 = SUM (of that whole thing) = 1/n^2+1/n^2+...+1/n^2=(n-1)/n^2

    Now, we take the limit as n --> infinity.
    This clearly goes to zero.
    Thus, the quadradic variation of f(t)=t of 1 (meaning on interval [0,1]) is zero.

    Now, they mention an interesting theorem.
    If f(t) is differenciable, then its quadradic variation is zero.
    Precisely what we got, zero.

    I hope this helps somehow.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Nov 2006
    Posts
    136
    yeah man thanks alot that helped alot!

    much appreciated
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Quadratic help
    Posted in the Algebra Forum
    Replies: 1
    Last Post: April 22nd 2010, 03:15 PM
  2. Quadratic help
    Posted in the Algebra Forum
    Replies: 2
    Last Post: April 7th 2010, 07:12 PM
  3. Looking for a Quadratic tip(s)
    Posted in the Algebra Forum
    Replies: 3
    Last Post: October 31st 2009, 04:45 AM
  4. Replies: 10
    Last Post: May 6th 2009, 10:52 AM
  5. Replies: 1
    Last Post: June 12th 2008, 10:30 PM

Search Tags


/mathhelpforum @mathhelpforum