P-variation explained

In mathematical analysis, p-variation is a collection of seminorms on functions from an ordered set to a metric space, indexed by a real number

p\geq1

. p-variation is a measure of the regularity or smoothness of a function. Specifically, if

f:I\to(M,d)

, where

(M,d)

is a metric space and I a totally ordered set, its p-variation is

\|f\|p-var=\left(\supD\sum

tk\inD

d(f(tk),f(tk-1))p\right)1/p

where D ranges over all finite partitions of the interval 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\circf

has finite
p
\alpha
-variation.

The case when p is one is called total variation, and functions with a finite 1-variation are called bounded variation functions.

Link with Hölder norm

One can interpret the p-variation as a parameter-independent version of the Hölder norm, which also extends to discontinuous functions.

If f is α - Hölder continuous (i.e. its α - Hölder norm is finite) then its

1{\alpha}
-variation is finite. Specifically, on an interval [''a'',''b''],

\|f

\|
1\alpha-var

\le\|f\|\alpha(b-a)\alpha

.

Conversely, if f is continuous and has finite p-variation, there exists a reparameterisation,

\tau

, such that

f\circ\tau

is

1/p-

Hölder continuous.[1]

If p is less than q then the space of functions of finite p-variation on a compact set is continuously embedded with norm 1 into those of finite q-variation. I.e.

\|f\|q-var\le\|f\|p-var

. However unlike the analogous situation with Hölder spaces the embedding is not compact. For example, consider the real functions on [0,1] given by
n
f
n(x)=x
. They are uniformly bounded in 1-variation and converge pointwise to a discontinuous function f but this not only is not a convergence in p-variation for any p but also is not uniform convergence.

Application to Riemann–Stieltjes integration

If f and g are functions from [''a'', ''b''] to

R

with no common discontinuities and with f having finite p-variation and g having finite q-variation, with
1p+1q>1
then the Riemann–Stieltjes Integral
b
\int
a

f(x)dg(x):=\lim|D|\to

\sum
tk\inD

f(tk)[g(tk+1)-g({tk})]

is well-defined. This integral is known as the Young integral because it comes from .[2] The value of this definite integral is bounded by the Young-Loève estimate as follows
b
\left|\int
a

f(x)dg(x)-f(\xi)[g(b)-g(a)]\right|\leC\|f\|p-var\|g\|q-var

where C is a constant which only depends on p and q and ξ is any number between a and b.[3] If f and g are continuous, the indefinite integral
w
F(w)=\int
a

f(x)dg(x)

is a continuous function with finite q-variation: If astb then

\|F\|q-var;[s,t]

, its q-variation on [''s'',''t''], is bounded by

C\|g\|q-var;[s,t](\|f\|p-var;[s,t]+\|f\|infty;[s,t])\le2C\|g\|q-var;[s,t](\|f\|p-var;[a,b]+f(a))

where C is a constant which only depends on p and q.[4]

Differential equations driven by signals of finite p-variation, p < 2

A function from

Rd

to e × d real matrices is called an

Re

-valued one-form on

Rd

.

If f is a Lipschitz continuous

Re

-valued one-form on

Rd

, and X is a continuous function from the interval [''a'',&nbsp;''b''] to

Rd

with finite p-variation with p less than 2, then the integral of f on X,
b
\int
a

f(X(t))dX(t)

, can be calculated because each component of f(X(t)) will be a path of finite p-variation and the integral is a sum of finitely many Young integrals. It provides the solution to the equation

dY=f(X)dX

driven by the path X.

More significantly, if f is a Lipschitz continuous

Re

-valued one-form on

Re

, and X is a continuous function from the interval [''a'',&nbsp;''b''] to

Rd

with finite p-variation with p less than 2, then Young integration is enough to establish the solution of the equation

dY=f(Y)dX

driven by the path X.[5]

Differential equations driven by signals of finite p-variation, p ≥ 2

The theory of rough paths generalises the Young integral and Young differential equations and makes heavy use of the concept of p-variation.

For Brownian motion

p-variation should be contrasted with the quadratic variation which is used in stochastic analysis, which takes one stochastic process to another. In particular the definition of quadratic variation looks a bit like the definition of p-variation, when p has the value 2. Quadratic variation is defined as a limit as the partition gets finer, whereas p-variation is a supremum over all partitions. Thus the quadratic variation of a process could be smaller than its 2-variation. If Wt is a standard Brownian motion on [0,&nbsp;''T''], then with probability one its p-variation is infinite for

p\le2

and finite otherwise. The quadratic variation of W is

[W]T=T

.

Computation of p-variation for discrete time series

For a discrete time series of observations X0,...,XN it is straightforward to compute its p-variation with complexity of O(N2). Here is an example C++ code using dynamic programming:double p_var(const std::vector& X, double p) There exist much more efficient, but also more complicated, algorithms for

R

-valued processes[6] [7] and for processes in arbitrary metric spaces.[7]

References

External links

Notes and References

  1. Web site: Ullrich. David C.. 27 Feb 2018. real analysis - Link between -variation and Hölder norm. 2021-07-02. Mathematics Stack Exchange.
  2. Web site: Lecture 7. Young's integral. 25 December 2012.
  3. Book: Friz. Peter K.. Victoir. Nicolas. Peter Friz. Multidimensional Stochastic Processes as Rough Paths: Theory and Applications. 2010. Cambridge University Press. Cambridge Studies in Advanced Mathematics.
  4. Book: Lyons. Terry. Caruana. Michael. Levy. Thierry. Differential equations driven by rough paths, vol. 1908 of Lecture Notes in Mathematics. 2007. Springer.
  5. Web site: Lecture 8. Young's differential equations. 26 December 2012.
  6. Butkus . V. . Norvaiša . R. . 2018 . Computation of p-variation . Lithuanian Mathematical Journal . 58 . 4 . 360–378 . 10.1007/s10986-018-9414-3 . 126246235 .
  7. Web site: P-var. . 8 May 2020.