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 [math]\displaystyle{ p\geq 1 }[/math]. p-variation is a measure of the regularity or smoothness of a function. Specifically, if [math]\displaystyle{ f:I\to(M,d) }[/math], where [math]\displaystyle{ (M,d) }[/math] is a metric space and I a totally ordered set, its p-variation is

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

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

Conversely, if f is continuous and has finite p-variation, there exists a reparameterisation, [math]\displaystyle{ \tau }[/math], such that [math]\displaystyle{ f\circ\tau }[/math] is [math]\displaystyle{ 1/p- }[/math]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. [math]\displaystyle{ \|f\|_{q\text{-var}}\le \|f\|_{p\text{-var}} }[/math]. However unlike the analogous situation with Hölder spaces the embedding is not compact. For example, consider the real functions on [0,1] given by [math]\displaystyle{ f_n(x)=x^n }[/math]. 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 [math]\displaystyle{ \frac1p+\frac1q\gt 1 }[/math] then the Riemann–Stieltjes Integral

[math]\displaystyle{ \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})] }[/math]

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

[math]\displaystyle{ \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}} }[/math]

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 [math]\displaystyle{ F(w)=\int_a^w f(x) \, dg(x) }[/math] is a continuous function with finite q-variation: If a ≤ s ≤ t ≤ b then [math]\displaystyle{ \|F\|_{q\text{-var};[s,t]} }[/math], its q-variation on [s,t], is bounded by [math]\displaystyle{ 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)) }[/math] 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, [math]\displaystyle{ \int_a^b f(X(t))\,dX(t) }[/math], 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 [math]\displaystyle{ dY=f(X)\,dX }[/math] 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 [math]\displaystyle{ dY=f(Y)\,dX }[/math] driven by the path X.

Differential equations driven by signals of finite p-variation, p [math]\displaystyle{ \ge }[/math] 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 [math]\displaystyle{ p\le2 }[/math] and finite otherwise. The quadratic variation of W is [math]\displaystyle{ [W]_T=T }[/math].

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