Thread: what is the finite variation function?

1. what is the finite variation function?

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

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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

2. Originally Posted by chogo
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:

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

Then we have a continuous function:

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

and a discontinuous function

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

such that:

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

RonL

3. Originally Posted by chogo