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 intervalI. 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 then its -variation is finite. Specifically, on an interval ,. Conversely, if f is continuous and has finite p-variation, there exists a reparameterisation,, such that is 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. . 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.
If f and g are functions from to ℝ 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.
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 to ℝd 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 ℝe-valued one-form on ℝe, and X is a continuous function from the interval to ℝd 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.
Rough differential equations
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, where it 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 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. Here is an example C++ code using dynamic programming: double p_var
There exist much more efficient, but also more complicated, algorithms for ℝ-valued processes and for processes in arbitrary metric spaces.
