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 $p\ge 1$. 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

$\Vert f{\Vert }_{p\text{-var}}={(\underset{D}{\sup }\sum _{t_k\in D}d(f(t_k),f(t_{k-1}){)}^p)}^{1/p}$

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 $\frac{1}{\alpha }$-variation is finite. Specifically, on an interval [a,b], $\Vert f{\Vert }_{\frac{1}{\alpha }\text{-var}}\le \Vert f{\Vert }_{\alpha }(b-a{)}^{\alpha }$.

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. $\Vert f{\Vert }_{q\text{-var}}\le \Vert f{\Vert }_{p\text{-var}}$. 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 ℝ with no common discontinuities and with f having finite p-variation and g having finite q-variation, with $\frac{1}{p}+\frac{1}{q}>1$ then the Riemann–Stieltjes Integral

$\underset{a}{\overset{b}{\int }}f(x)dg(x):=\underset{|D|\to 0}{\lim }\sum _{t_k\in D}f(t_k)[g(t_{k+1})-g(t_k)]$

is well-defined. This integral is known as the Young integral because it comes from (Young 1936). The value of this definite integral is bounded by the Young-Loève estimate as follows

$|\underset{a}{\overset{b}{\int }}f(x)dg(x)-f(\xi )[g(b)-g(a)]|\le C\Vert f{\Vert }_{p\text{-var}}\Vert g{\Vert }_{q\text{-var}}$

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 $F(w)=\underset{a}{\overset{w}{\int }}f(x)dg(x)$ is a continuous function with finite q-variation: If a ≤ s ≤ t ≤ b then $\Vert F{\Vert }_{q\text{-var};[s,t]}$, its q-variation on [s,t], is bounded by $C\Vert g{\Vert }_{q\text{-var};[s,t]}(\Vert f{\Vert }_{p\text{-var};[s,t]}+\Vert f{\Vert }_{\mathrm{\infty };[s,t]})\le 2C\Vert g{\Vert }_{q\text{-var};[s,t]}(\Vert f{\Vert }_{p\text{-var};[a,b]}+f(a))$ 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 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, $\underset{a}{\overset{b}{\int }}f(X(t))dX(t)$, 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 [0, T], then with probability one its p-variation is infinite for $p\le 2$ 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,...,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<double>& X, double p) {
	if (X.size() == 0)
		return 0.0;
	std::vector<double> cum_p_var(X.size(), 0.0);   // cumulative p-variation
	for (size_t n = 1; n < X.size(); n++) {
		for (size_t k = 0; k < n; k++) {
			cum_p_var[n] = std::max(cum_p_var[n], cum_p_var[k] + std::pow(std::abs(X[n] - X[k]), p));
		}
	}
	return std::pow(cum_p_var.back(), 1./p);
}

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

References

• Young, L.C. (1936), "An inequality of the Hölder type, connected with Stieltjes integration", Acta Mathematica 67 (1): 251–282, doi:10.1007/bf02401743 .

External links

• Continuous Paths with bounded p-variation Fabrice Baudoin
• On the Young integral, truncated variation and rough paths Rafał M. Łochowski