p-variation

<h2 id="link-with-hlder-norm">Link with Hölder norm</h2>
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–<a href="/facts/H%C3%B6lder_continuous/CI0yp6kD">Hölder continuous</a> (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.

<h2 id="application-to-riemannstieltjes-integration">Application to Riemann–Stieltjes integration</h2>
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 <a href="/facts/Riemann%E2%80%93Stieltjes_Integral/C08wyfWn">Riemann–Stieltjes Integral</a>

∫
          
            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).<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> 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.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>
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.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

<h2 id="differential-equations-driven-by-signals-of-finite-p-variation-p-2">Differential equations driven by signals of finite p-variation, p < 2</h2>
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.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

<h2 id="differential-equations-driven-by-signals-of-finite-p-variation-p-2">Differential equations driven by signals of finite p-variation, p ≥ 2</h2>
The theory of <a href="/facts/Rough_path/oL0RfX4W">rough paths</a> generalises the Young integral and Young differential equations and makes heavy use of the concept of p-variation.

<h2 id="for-brownian-motion">For Brownian motion</h2>
p-variation should be contrasted with the <a href="/facts/Quadratic_variation/15jHz4xU">quadratic variation</a> which is used in <a href="/facts/Stochastic_analysis/QgY91tp2">stochastic analysis</a>, 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 <a href="/facts/Wiener_process/KPh7vYd7">Brownian motion</a> 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}
 
.

<h2 id="computation-of-p-variation-for-discrete-time-series">Computation of p-variation for discrete time series</h2>
For a discrete time series of observations X0,...,XN it is straightforward to compute its p-variation with complexity of <a href="/facts/Big-O_notation/weFFjSWg">O</a>(N2). Here is an example C++ code using <a href="/facts/Dynamic_programming/CLcabtmk">dynamic programming</a>:

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 processes<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>
<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>
and for processes in arbitrary metric spaces.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>

<ul><li>Young, L.C. (1936), "An inequality of the Hölder type, connected with Stieltjes integration", Acta Mathematica, 67 (1): 251–282, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fbf02401743">10.1007/bf02401743</a>.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="https://fabricebaudoin.wordpress.com/2012/12/24/lecture-6-continuous-paths-with-bounded-p-variation/">Continuous Paths with bounded p-variation</a> Fabrice Baudoin</li>
<li><a href="http://web.sgh.waw.pl/~rlocho/UCT_talk.pdf">On the Young integral, truncated variation and rough paths</a> Rafał M. Łochowski</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn: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. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">"Lecture 7. Young's integral". 25 December 2012. <a href="https://fabricebaudoin.wordpress.com/2012/12/25/lecture-7-youngs-integral/" target="_blank">https://fabricebaudoin.wordpress.com/2012/12/25/lecture-7-youngs-integral/</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="/wiki/Peter_Friz" target="_blank">/wiki/Peter_Friz</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Lyons, Terry; Caruana, Michael; Levy, Thierry (2007). Differential equations driven by rough paths, vol. 1908 of Lecture Notes in Mathematics. Springer. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">"Lecture 8. Young's differential equations". 26 December 2012. <a href="https://fabricebaudoin.wordpress.com/2012/12/26/lecture-8-youngs-differential-equations/" target="_blank">https://fabricebaudoin.wordpress.com/2012/12/26/lecture-8-youngs-differential-equations/</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">"P-var". GitHub. 8 May 2020. <a href="https://github.com/khumarahn/p-var" target="_blank">https://github.com/khumarahn/p-var</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">"P-var". GitHub. 8 May 2020. <a href="https://github.com/khumarahn/p-var" target="_blank">https://github.com/khumarahn/p-var</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
</ol>

p-variation open-in-new

p-variation