P-variation

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 $p\geq 1$. ''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

:$\, f \, _ = \left(\sup_D\sum_d(f(t_k),f(t_))^p\right)^$

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 $g\circ f$ has finite $\frac$-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 $\frac1$-variation is finite. Specifically, on an interval 'a'',''b'' $\, f \, _\le \, f \, _(b-a)^\alpha$.

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.

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.
$\, f\, _\le \, f\, _$. However unlike the analogous situation with Hölder spaces the embedding is not compact.
For example, consider the real functions on ,1given by $f_n(x)=x^n$. 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 $\frac1p+\frac1q>1$ then the Riemann–Stieltjes Integral
:$\int_a^b f(x) \, dg(x):=\lim_\sum_f(t_k)$(t_)-g()/math>
is well-defined. This integral is known as the ''Young integral'' because it comes from . The value of this definite integral is bounded by the Young-Loève estimate as follows
:\left, \int_a^b f(x) \, dg(x)-f(\xi) (b)-g(a)\le C\,\, f\, _\, \,g\, _

where ''C'' is a constant which only depends on ''p'' and ''q'' and ξ is any number between ''a'' and ''b''.
If ''f'' and ''g'' are continuous, the indefinite integral $F(w)=\int_a^w f(x) \, dg(x)$ is a continuous function with finite ''q''-variation: If ''a'' ≤ ''s'' ≤ ''t'' ≤ ''b'' then $\, F\, _$, its ''q''-variation on 's'',''t'' is bounded by
$C\, g\, _(\, f\, _+\, f\, _)\le2C\, g\, _(\, f\, _+f(a))$
where ''C'' is a constant which only depends on ''p'' and ''q''.

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

A function from ℝ^''d'' to ''e'' × ''d'' real matrices is called an ℝ^''e''-valued one-form on ℝ^''d''.

If ''f'' is a Lipschitz continuous ℝ^''e''-valued one-form on ℝ^''d'', and ''X'' is a continuous function from the interval 'a'', ''b''to ℝ^''d'' with finite ''p''-variation with ''p'' less than 2, then the integral of ''f'' on ''X'', $\int_a^b f(X(t))\,dX(t)$, 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 ℝ^''e''-valued one-form on ℝ^''e'', and ''X'' is a continuous function from the interval 'a'', ''b''to ℝ^''d'' 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''.

Differential equations driven by signals of finite ''p''-variation, ''p'' $\ge$ 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 ''W''_''t'' is a standard Brownian motion on , ''T'' then with probability one its ''p''-variation is infinite for $p\le2$ and finite otherwise. The quadratic variation of ''W'' is T=T.

Computation of ''p''-variation for discrete time series

For a discrete time series of observations ''X₀,...,X_N'' 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

and for processes in arbitrary metric spaces.

References

*{{citation , last=Young, first=L.C., title=An inequality of the Hölder type, connected with Stieltjes integration, journal=Acta Mathematica, volume=67, year=1936, issue=1, pages=251–282, doi=10.1007/bf02401743, doi-access=free.

External links

Continuous Paths with bounded p-variation
Fabrice Baudoin

On the Young integral, truncated variation and rough paths
Rafał M. Łochowski

Mathematical analysis