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.
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
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
For example for p=2, this corresponds to the concept of quadratic variation, which is different from 2-variation.
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
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.
However unlike the analogous situation with Hölder spaces the embedding is not compact.
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.
[2] 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.
[3] If f and g are continuous, the indefinite integral
is a continuous function with finite q-variation: If a ≤ s ≤ t ≤ b then
to e × d real matrices is called an
, 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.
More significantly, if f is a Lipschitz continuous
, 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
[5] The theory of rough paths generalises the Young integral and Young differential equations and makes heavy use of the concept of p-variation.
p-variation should be contrasted with the quadratic variation which is used in stochastic analysis, which takes one stochastic process to another.
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
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: There exist much more efficient, but also more complicated, algorithms for