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-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 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 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.
This concept should not be confused with the notion of p-th variation along a sequence of partitions, which is computed as a limit along a given sequence of time partitions:[1]
For example for p=2, this corresponds to the concept of quadratic variation, which is different from 2-variation.
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 -variation is finite. Specifically, on an interval [a,b], .
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 [0,1] 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.
Application to Riemann–Stieltjes integration
If f and g are functions from [a, b] to 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 Young (1936).[2] 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.[3] 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 [s,t], is bounded by 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 to e × d real matrices is called an -valued one-form on .
If f is a Lipschitz continuous -valued one-form on , and X is a continuous function from the interval [a, b] to with finite p-variation with p less than 2, then the integral of f on X, , 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 driven by the path X.
More significantly, if f is a Lipschitz continuous -valued one-form on , and X is a continuous function from the interval [a, b] to 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.[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 Wt is a standard Brownian motion on [0, T], 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(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 -valued processes[6] [7] and for processes in arbitrary metric spaces.[7]
References
- ^ Cont, R.; Perkowski, N. (2019). "Pathwise integration and change of variable formulas for continuous paths with arbitrary regularity". Transactions of the American Mathematical Society. 6: 161–186. arXiv:1803.09269. doi:10.1090/btran/34.
- ^ "Lecture 7. Young's integral". 25 December 2012.
- ^ Friz, Peter K.; Victoir, Nicolas (2010). Multidimensional Stochastic Processes as Rough Paths: Theory and Applications (Cambridge Studies in Advanced Mathematics ed.). Cambridge University Press.
- ^ Lyons, Terry; Caruana, Michael; Levy, Thierry (2007). Differential equations driven by rough paths, vol. 1908 of Lecture Notes in Mathematics. Springer.
- ^ "Lecture 8. Young's differential equations". 26 December 2012.
- ^ 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.
- ^ a b "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