p-variation

In <a href="/facts/Mathematical_analysis/XXeR26Ih">mathematical analysis</a>, p-variation is a collection of <a href="/facts/Norm_(mathematics)/xIbR4uE1">seminorms</a> on functions from an ordered set to a <a href="/facts/Metric_space/gnBBRXtc">metric space</a>, indexed by a real number 
 
 
 
 p
 ≥
 1
 
 
 {\displaystyle p\geq 1}
 
. p-variation is a measure of the regularity or smoothness of a function. Specifically, if 
 
 
 
 f
 :
 I
 →
 (
 M
 ,
 d
 )
 
 
 {\displaystyle f:I\to (M,d)}
 
, where 
 
 
 
 (
 M
 ,
 d
 )
 
 
 {\displaystyle (M,d)}
 
 is a metric space and I a totally ordered set, its p-variation is:

 
 
 
 ‖
 f
 
 ‖
 
 p
 
 -var
 
 
 
 =
 
 
 (
 
 
 sup
 
 D
 
 
 
 ∑
 
 
 t
 
 k
 
 
 ∈
 D
 
 
 d
 (
 f
 (
 
 t
 
 k
 
 
 )
 ,
 f
 (
 
 t
 
 k
 −
 1
 
 
 )
 
 )
 
 p
 
 
 
 )
 
 
 1
 
 /
 
 p
 
 
 
 
 {\displaystyle \|f\|_{p{\text{-var}}}=\left(\sup _{D}\sum _{t_{k}\in D}d(f(t_{k}),f(t_{k-1}))^{p}\right)^{1/p}}

where D ranges over all finite <a href="/facts/Partition_of_an_interval/FKoPazKA">partitions of the interval</a> I.
The p variation of a function decreases with p. If f has finite p-variation and g is an α-Hölder continuous function, then 
 
 
 
 g
 ∘
 f
 
 
 {\displaystyle g\circ f}
 
 has finite 
 
 
 
 
 
 p
 α
 
 
 
 
 {\displaystyle {\frac {p}{\alpha }}}
 
-variation.
The case when p is one is called <a href="/facts/Total_variation/tyY5fTz9">total variation</a>, and functions with a finite 1-variation are called <a href="/facts/Bounded_variation/tIjQTpjh">bounded variation</a> functions.
This concept should not be confused with the notion of p-th variation along a sequence of partitions, which is computed as a limit along a given sequence 
 
 
 
 (
 
 D
 
 n
 
 
 )
 
 
 {\displaystyle (D_{n})}
 
 of time partitions:

 
 
 
 [
 f
 
 ]
 
 p
 
 
 =
 
 (
 
 
 lim
 
 n
 →
 ∞
 
 
 
 ∑
 
 
 t
 
 k
 
 
 n
 
 
 ∈
 
 D
 
 n
 
 
 
 
 d
 (
 f
 (
 
 t
 
 k
 
 
 n
 
 
 )
 ,
 f
 (
 
 t
 
 k
 −
 1
 
 
 n
 
 
 )
 
 )
 
 p
 
 
 
 )
 
 
 
 {\displaystyle [f]_{p}=\left(\lim _{n\to \infty }\sum _{t_{k}^{n}\in D_{n}}d(f(t_{k}^{n}),f(t_{k-1}^{n}))^{p}\right)}

For example for p=2, this corresponds to the concept of quadratic variation, which is different from 2-variation.

p-variation open-in-new

p-variation