P-variation (original) (raw)

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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} {\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)} {\displaystyle f:I\to (M,d)}, where ( M , d ) {\displaystyle (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}} {\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} {\displaystyle g\circ f} has finite p α {\displaystyle {\frac {p}{\alpha }}} {\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})} {\displaystyle (D_{n})} of time partitions:[1]

[ 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)} {\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.

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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 }}} {\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 }} {\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}}}} {\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}} {\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

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If f and g are functions from [a, _b_] to R {\displaystyle \mathbb {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} {\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}})]} {\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}}}} {\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)} {\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]}} ![{\displaystyle |F|{q{\text{-var}};[s,t]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29de4a2c81ecbc20d8b5d48e4509323e15d8a9ac), 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))} ![{\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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bd454c6633a99ebb81edfd014b9169d3ffe9a14)where C is a constant which only depends on p and q.[4]

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

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

If f is a Lipschitz continuous R e {\displaystyle \mathbb {R} ^{e}} {\displaystyle \mathbb {R} ^{e}}-valued one-form on R d {\displaystyle \mathbb {R} ^{d}} {\displaystyle \mathbb {R} ^{d}}, and X is a continuous function from the interval [a, _b_] to R d {\displaystyle \mathbb {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)} {\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} {\displaystyle dY=f(X)\,dX} driven by the path X.

More significantly, if f is a Lipschitz continuous R e {\displaystyle \mathbb {R} ^{e}} {\displaystyle \mathbb {R} ^{e}}-valued one-form on R e {\displaystyle \mathbb {R} ^{e}} {\displaystyle \mathbb {R} ^{e}}, and X is a continuous function from the interval [a, _b_] to R d {\displaystyle \mathbb {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} {\displaystyle dY=f(Y)\,dX} driven by the path X.[5]

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

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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

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_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 ≤ 2 {\displaystyle p\leq 2} {\displaystyle p\leq 2} and finite otherwise. The quadratic variation of W is [ W ] T = T {\displaystyle [W]_{T}=T} ![{\displaystyle [W]{T}=T}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8af2af4c43ea2fe7421d314221d4549e22d9007f).

Computation of _p_-variation for discrete time series

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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& X, double p) { if (X.size() == 0) return 0.0; std::vector 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} } {\displaystyle \mathbb {R} }-valued processes[6] [7]and for processes in arbitrary metric spaces.[7]

  1. ^ 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.
  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. ^ a b "P-var". GitHub. 8 May 2020.