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Math Help - what is the finite variation function?

  1. #1
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    what is the finite variation function?

    I am learning stochastic process for my research. However i am confused with the whole continuous - discontinuous thing.

    The book im reading refers to something called the finite variation function, but i cant find much info on it on google. Does anyone know about this?

    __________________________________________________ _________________

    Also there is a small derivation

    if we define a discontinuous part gd of g as gd(t)

    and the continuous part gc or g by gc(t) = g(t) - gd(t). It is clear gd only changes by jumps.

    What is meant by this?

    thank you for any help, this forum is a gold mine
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  2. #2
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    Quote Originally Posted by chogo View Post
    Also there is a small derivation

    if we define a discontinuous part gd of g as gd(t)

    and the continuous part gc or g by gc(t) = g(t) - gd(t). It is clear gd only changes by jumps.

    What is meant by this?
    Its means you can decompose an function with jump discontiuities (nicely behaved
    possibly) into a continuous function and a piecewise constant function that
    includes all of the jumps.

    For example consider the function:

    <br />
f(x)=\left\{ \begin {array} {cc}x^2,& \ \ \ ...\ x<0\\1+x, & \ \ \ ...\ x \ge 0 \end{array} \right.<br />

    Then we have a continuous function:

    <br />
fc(x)=\left\{ \begin {array} {cc}x^2,& \ \ \ ...\ x<0\\x, & \ \ \ ...\ x \ge 0 \end{array} \right.<br />

    and a discontinuous function

    <br />
fd(x)=\left\{ \begin {array} {cc}0,& \ \ \ ...\ x<0\\1, & \ \ \ ...\ x \ge 0 \end{array} \right.<br />

    such that:

    f(x)=fc(x)+fd(x)

    RonL
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by chogo View Post
    The book im reading refers to something called the finite variation function, but i cant find much info on it on google. Does anyone know about this?
    I suspect its another name for, or related to the notion of, a function of bounded variation. If so see here.

    RonL
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