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-variation is a measure of the regularity or smoothness of a function. Specifically, if
, where
is a metric space and
I a totally ordered set, its
p-variation is
\|f\|p-var=\left(\supD\sum
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
has finite
-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
-variation is finite. Specifically, on an interval [''a'',''b''],
\|f
\le\|f\|\alpha(b-a)\alpha
.
Conversely, if f is continuous and has finite p-variation, there exists a reparameterisation,
, such that
is
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.
. However unlike the analogous situation with Hölder spaces the embedding is not compact. For example, consider the real functions on [0,1] given by
. 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
with no common discontinuities and with
f having finite
p-variation and
g having finite
q-variation, with
then the
Riemann–Stieltjes Integral
f(x)dg(x):=\lim|D|\to
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
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
is a continuous function with finite
q-variation: If
a ≤
s ≤
t ≤
b then
, 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
to
e ×
d real matrices is called an
-valued one-form on
.
If f is a Lipschitz continuous
-valued one-form on
, and
X is a continuous function from the interval [''a'', ''b''] to
with finite
p-variation with
p less than 2, then the integral of
f on
X,
, 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
driven by the path
X.
More significantly, if f is a Lipschitz continuous
-valued one-form on
, and
X is a continuous function from the interval [''a'', ''b''] to
with finite
p-variation with
p less than 2, then Young integration is enough to establish the solution of the equation
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, ''T''], then with probability one its p-variation is infinite for
and finite otherwise. The quadratic variation of
W is
.
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
-valued processes
[6] [7] and for processes in arbitrary metric spaces.
[7] References
External links
Notes and References
- Web site: Ullrich. David C.. 27 Feb 2018. real analysis - Link between -variation and Hölder norm. 2021-07-02. Mathematics Stack Exchange.
- Web site: Lecture 7. Young's integral. 25 December 2012.
- 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.
- Book: Lyons. Terry. Caruana. Michael. Levy. Thierry. Differential equations driven by rough paths, vol. 1908 of Lecture Notes in Mathematics. 2007. Springer.
- Web site: Lecture 8. Young's differential equations. 26 December 2012.
- Butkus . V. . Norvaiša . R. . 2018 . Computation of p-variation . Lithuanian Mathematical Journal . 58 . 4 . 360–378 . 10.1007/s10986-018-9414-3 . 126246235 .
- Web site: P-var. . 8 May 2020.