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


 * $$ \| f \|_{p\text{-var}} = \left(\sup_D\sum_{t_k\in D}d(f(t_k),f(t_{k-1}))^p\right)^{1/p}$$

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 $$g\circ f$$ has finite $$\frac{p}{\alpha}$$-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 $$\frac1{\alpha}$$-variation is finite. Specifically, on an interval [a,b], $$\| f \|_{\frac1\alpha\text{-var}}\le \| f \|_{\alpha}(b-a)^\alpha$$.

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. $$\|f\|_{q\text{-var}}\le \|f\|_{p\text{-var}}$$. However unlike the analogous situation with Hölder spaces the embedding is not compact. For example, consider the real functions on [0,1] given 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 $$\mathbb{R}$$ 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
 * $$\int_a^b f(x) \, dg(x):=\lim_{|D|\to 0}\sum_{t_k\in D}f(t_k)[g(t_{k+1})-g({t_k})]$$

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
 * $$\left|\int_a^b f(x) \, dg(x)-f(\xi)[g(b)-g(a)]\right|\le C\,\|f\|_{p\text{-var}}\|\,g\|_{q\text{-var}}$$

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 $$\|F\|_{q\text{-var};[s,t]}$$, its q-variation on [s,t], is bounded by $$C\|g\|_{q\text{-var};[s,t]}(\|f\|_{p\text{-var};[s,t]}+\|f\|_{\infty;[s,t]})\le2C\|g\|_{q\text{-var};[s,t]}(\|f\|_{p\text{-var};[a,b]}+f(a))$$ 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 $$\mathbb{R}^{d}$$ to e × d real matrices is called an $$\mathbb{R}^{e}$$-valued one-form on $$\mathbb{R}^{d}$$.

If f is a Lipschitz continuous $$\mathbb{R}^{e}$$-valued one-form on $$\mathbb{R}^{d}$$, and X is a continuous function from the interval [a, b] to $$\mathbb{R}^{d}$$ 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 $$\mathbb{R}^{e}$$-valued one-form on $$\mathbb{R}^{e}$$, and X is a continuous function from the interval [a, b] to $$\mathbb{R}^{d}$$ 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 ≥ 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 $$p\le2$$ and finite otherwise. The quadratic variation of W is $$[W]_T=T$$.

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: There exist much more efficient, but also more complicated, algorithms for $$\mathbb{R}$$-valued processes and for processes in arbitrary metric spaces.