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 ≥ 1 {\displaystyle p\geq 1} . p-variation is a measure of the regularity or smoothness of a function. Specifically, if f : I → ( M , d ) {\displaystyle f:I\to (M,d)} , where ( M , d ) {\displaystyle (M,d)} is a metric space and I a totally ordered set, its p-variation is:
‖ f ‖ p -var = ( sup D ∑ t k ∈ D d ( f ( t k ) , f ( t k − 1 ) ) p ) 1 / p {\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 g ∘ f {\displaystyle g\circ f} has finite p α {\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.
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 ( D n ) {\displaystyle (D_{n})} of time partitions:
[ f ] p = ( lim n → ∞ ∑ t k n ∈ D n d ( f ( t k n ) , f ( t k − 1 n ) ) p ) {\displaystyle [f]_{p}=\left(\lim _{n\to \infty }\sum _{t_{k}^{n}\in D_{n}}d(f(t_{k}^{n}),f(t_{k-1}^{n}))^{p}\right)}
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 1 α {\displaystyle {\frac {1}{\alpha }}} -variation is finite. Specifically, on an interval [a,b], ‖ f ‖ 1 α -var ≤ ‖ f ‖ α ( b − a ) α {\displaystyle \|f\|_{{\frac {1}{\alpha }}{\text{-var}}}\leq \|f\|_{\alpha }(b-a)^{\alpha }} .
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 -var ≤ ‖ f ‖ p -var {\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 f n ( x ) = x n {\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 R {\displaystyle \mathbb {R} } with no common discontinuities and with f having finite p-variation and g having finite q-variation, with 1 p + 1 q > 1 {\displaystyle {\frac {1}{p}}+{\frac {1}{q}}>1} then the Riemann–Stieltjes Integral
∫ a b f ( x ) d g ( x ) := lim | D | → 0 ∑ t k ∈ D f ( t k ) [ g ( t k + 1 ) − g ( t k ) ] {\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
| ∫ a b f ( x ) d g ( x ) − f ( ξ ) [ g ( b ) − g ( a ) ] | ≤ C ‖ f ‖ p -var ‖ g ‖ q -var {\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 F ( w ) = ∫ a w f ( x ) d g ( x ) {\displaystyle 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 -var ; [ s , t ] {\displaystyle \|F\|_{q{\text{-var}};[s,t]}} , its q-variation on [s,t], is bounded by C ‖ g ‖ q -var ; [ s , t ] ( ‖ f ‖ p -var ; [ s , t ] + ‖ f ‖ ∞ ; [ s , t ] ) ≤ 2 C ‖ g ‖ q -var ; [ s , t ] ( ‖ f ‖ p -var ; [ a , b ] + f ( a ) ) {\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 R d {\displaystyle \mathbb {R} ^{d}} to e × d real matrices is called an R e {\displaystyle \mathbb {R} ^{e}} -valued one-form on R d {\displaystyle \mathbb {R} ^{d}} .
If f is a Lipschitz continuous R e {\displaystyle \mathbb {R} ^{e}} -valued one-form on R d {\displaystyle \mathbb {R} ^{d}} , and X is a continuous function from the interval [a, b] to R d {\displaystyle \mathbb {R} ^{d}} with finite p-variation with p less than 2, then the integral of f on X, ∫ a b f ( X ( t ) ) d X ( t ) {\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 d Y = f ( X ) d X {\displaystyle dY=f(X)\,dX} driven by the path X.
More significantly, if f is a Lipschitz continuous R e {\displaystyle \mathbb {R} ^{e}} -valued one-form on R e {\displaystyle \mathbb {R} ^{e}} , and X is a continuous function from the interval [a, b] to R d {\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 d Y = f ( Y ) d X {\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 Wt is a standard Brownian motion on [0, T], then with probability one its p-variation is infinite for p ≤ 2 {\displaystyle p\leq 2} and finite otherwise. The quadratic variation of W is [ W ] T = T {\displaystyle [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:
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 {\displaystyle \mathbb {R} } -valued processes6 7 and for processes in arbitrary metric spaces.8
- 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
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. /wiki/ArXiv_(identifier) ↩
"Lecture 7. Young's integral". 25 December 2012. https://fabricebaudoin.wordpress.com/2012/12/25/lecture-7-youngs-integral/ ↩
Friz, Peter K.; Victoir, Nicolas (2010). Multidimensional Stochastic Processes as Rough Paths: Theory and Applications (Cambridge Studies in Advanced Mathematics ed.). Cambridge University Press. /wiki/Peter_Friz ↩
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. https://fabricebaudoin.wordpress.com/2012/12/26/lecture-8-youngs-differential-equations/ ↩
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. /wiki/Doi_(identifier) ↩
"P-var". GitHub. 8 May 2020. https://github.com/khumarahn/p-var ↩
"P-var". GitHub. 8 May 2020. https://github.com/khumarahn/p-var ↩