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

${\displaystyle \|f\|_{p{\text{-var}}}=\left(\sup _{D}\sum _{t_{k}\in D}d(f(t_{k}),f(t_{k-1}))^{p}\right)^{1/p}}$

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 ${\displaystyle g\circ f}$ has finite ${\displaystyle {\frac {p}{\alpha }}}$-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 ${\displaystyle {\frac {1}{\alpha }}}$-variation is finite. Specifically, on an interval [a,b], ${\displaystyle \|f\|_{{\frac {1}{\alpha }}{\text{-var}}}\leq \|f\|_{\alpha }(b-a)^{\alpha }}$.

Conversely, if f is continuous and has finite p-variation, there exists a reparameterisation, ${\displaystyle \tau }$, such that ${\displaystyle f\circ \tau }$ is ${\displaystyle 1/p-}$Hölder continuous.^[1]

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. ${\displaystyle \|f\|_{q{\text{-var}}}\leq \|f\|_{p{\text{-var}}}}$. However unlike the analogous situation with Hölder spaces the embedding is not compact. For example, consider the real functions on [0,1] given by ${\displaystyle f_{n}(x)=x^{n}}$. 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 ${\displaystyle \mathbb {R} }$ with no common discontinuities and with f having finite p-variation and g having finite q-variation, with ${\displaystyle {\frac {1}{p}}+{\frac {1}{q}}>1}$ then the Riemann–Stieltjes Integral

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

is well-defined. This integral is known as the Young integral because it comes from Young (1936).^[2] The value of this definite integral is bounded by the Young-Loève estimate as follows

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

where C is a constant which only depends on p and q and ξ is any number between a and b.^[3]If f and g are continuous, the indefinite integral ${\displaystyle F(w)=\int _{a}^{w}f(x)\,dg(x)}$ is a continuous function with finite q-variation: If a ≤ s ≤ t ≤ b then ${\displaystyle \|F\|_{q{\text{-var}};[s,t]}}$, its q-variation on [s,t], is bounded by ${\displaystyle C\|g\|_{q{\text{-var}};[s,t]}(\|f\|_{p{\text{-var}};[s,t]}+\|f\|_{\infty ;[s,t]})\leq 2C\|g\|_{q{\text{-var}};[s,t]}(\|f\|_{p{\text{-var}};[a,b]}+f(a))}$where C is a constant which only depends on p and q.^[4]

Differential equations driven by signals of finite p-variation, p < 2

A function from ${\displaystyle \mathbb {R} ^{d}}$ to e × d real matrices is called an ${\displaystyle \mathbb {R} ^{e}}$-valued one-form on ${\displaystyle \mathbb {R} ^{d}}$.

If f is a Lipschitz continuous ${\displaystyle \mathbb {R} ^{e}}$-valued one-form on ${\displaystyle \mathbb {R} ^{d}}$, and X is a continuous function from the interval [a, b] to ${\displaystyle \mathbb {R} ^{d}}$ with finite p-variation with p less than 2, then the integral of f on X, ${\displaystyle \int _{a}^{b}f(X(t))\,dX(t)}$, 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 ${\displaystyle dY=f(X)\,dX}$ driven by the path X.

More significantly, if f is a Lipschitz continuous ${\displaystyle \mathbb {R} ^{e}}$-valued one-form on ${\displaystyle \mathbb {R} ^{e}}$, and X is a continuous function from the interval [a, b] to ${\displaystyle \mathbb {R} ^{d}}$ with finite p-variation with p less than 2, then Young integration is enough to establish the solution of the equation ${\displaystyle dY=f(Y)\,dX}$ driven by the path X.^[5]

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 W_t is a standard Brownian motion on [0, T], then with probability one its p-variation is infinite for ${\displaystyle p\leq 2}$ and finite otherwise. The quadratic variation of W is ${\displaystyle [W]_{T}=T}$.

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-variationfor (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 ${\displaystyle \mathbb {R} }$-valued processes^[6][7]and for processes in arbitrary metric spaces.^[7]

References

1. Ullrich, David C. (27 Feb 2018). "real analysis - Link between p-variation and Hölder norm". Mathematics Stack Exchange. Retrieved 2021-07-02.
2. "Lecture 7. Young's integral". 25 December 2012.
3. Friz, Peter K.; Victoir, Nicolas (2010). Multidimensional Stochastic Processes as Rough Paths: Theory and Applications (Cambridge Studies in Advanced Mathematics ed.). Cambridge University Press.
4. Lyons, Terry; Caruana, Michael; Levy, Thierry (2007). Differential equations driven by rough paths, vol. 1908 of Lecture Notes in Mathematics. Springer.
5. "Lecture 8. Young's differential equations". 26 December 2012.
6. Butkus, V.; Norvaiša, R. (2018). "Computation of p-variation". Lithuanian Mathematical Journal. 58 (4): 360–378. doi:10.1007/s10986-018-9414-3. S2CID 126246235.
7. "P-var". GitHub. 8 May 2020.

• 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