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

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 then its -variation is finite. Specifically, on an interval,.
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 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 to with no common discontinuities and with f having finite p-variation and g having finite q-variation, with then the Riemann–Stieltjes Integral
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
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 is a continuous function with finite q-variation: If a ≤ s ≤ t ≤ b then, its q-variation on, is bounded by
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 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 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 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 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.

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 W_t is a standard Brownian motion on, 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,...,X_N it is straightforward to compute its p-variation with complexity of O. Here is an example C++ code using dynamic programming:

double p_var

There exist much more efficient, but also more complicated, algorithms for -valued processes
and for processes in arbitrary metric spaces.