Use this tag for questions about $p$-variation norms, the study of finiteness of the $p$- variation of functions, which is a generalization of the total variation.
Questions tagged [p-variation]
19 questions
5
votes
1 answer
A function with cubic variation
Let $W(\omega,x):\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ be a Brownian motion where $\Omega$ is the sample space. Recall that the quadratic variation of $W$ over the interval $[a,b]$ equals $b-a$ almost surely, that is to say:
$$\langle W…
Mathew
- 1,928
4
votes
1 answer
Link between Riesz p-variation and absolute continuity
Fix $p\in(1,+\infty)$ and for $f:[0,1]\to\mathbb{R}$ define
$$
V_p(f)=\sup \sum_{i=0}^{n-1}\frac{|f(t_{i+1})-f(t_i)|^p}{(t_{i+1}-t_i)^{p-1}}
$$
where the sup is taken over all $n\in\mathbb{N}$ and among all possible partitions…
Motticoz
- 398
4
votes
1 answer
Why is $\lim_{n\to\infty} 2^n \Psi\left(\frac{r}{2^n}\right)=0$, for some specific function $\Psi$ defined in the question?
This comes from the proof of the first theorem in this blog article.
The paths $x:[0,T]\to\mathbb{R}^d$ and $y:[0,T]\to\mathbb{R}^{e\times d}$ are of bounded total variation. The real numbers $p,q>1$ are such that $\frac{1}{p}+\frac{1}{q}>1$.…
man_in_green_shirt
- 2,676
3
votes
0 answers
An example of functions of bounded p and q-variations, $\frac{1}{p} + \frac{1}{q} = 1$, for which the Riemann-Stieltjes integral does not exist
Let $f$, $g$: $[a, b] \to \mathbb{R}$, $\mathrm{Var}(p, f) < \infty$, $\mathrm{Var}(q, g) < \infty$, $\frac{1}{p} + \frac{1}{q} = 1$. Why then
$\int_{a}^{b} fdg$ maybe it is not defined? There were ideas to take partial sums of the Weierstrass…
3
votes
0 answers
Continuity and variation
Are there continuous functions having infinite $p$-variation for any $p\ge 1$?
I guess yes since for any $p=1,2,\dots$ and any $x\in \mathbb R$, there presumably is a function $f : [p-1,p] \rightarrow \mathbb R$ with $f(p-1)=x$ having infinite…
Kehrwert
- 669
2
votes
1 answer
Why does the definition of quadratic variation for stochastic processes use a vanishing mesh instead of a supremum like in p-variation?
I have a question regarding the definitions of quadratic variation for stochastic processes and p-variation for general functions.
Specifically, for a stochastic process $X_t$, the quadratic variation $[X]_T$ is defined using the limit as the mesh…
2
votes
0 answers
When does $p$-variation mesh size tend to $0$?
Let $u: [a,b] \to \mathbb{R}$ and define its $p$-Wiener variation as $$V_p(u) := \sup \left\{ \left(\sum_i |u(x_i)-u(x_{i-1})|^p\right)^{1/p}\right\}, $$ where the supremum ranges over finite partitions $P=\{x_1,\ldots, x_n\}$ of $[a,b]$.
For $p =…
nullUser
- 28,703
2
votes
1 answer
Link between $p-variation$ and $Holder$ norm
I'm studying some basic notions of Young Integration and I got stuck with the so called "Link between $p-variation$ and $Holder$ norm" found on the wikipedia page: the criminal page.
In particular it says that if $f$ has finite $p-variation$ and it…
JCF
- 683
1
vote
0 answers
Distribution of $p$-variation of Brownian motion
Given a path $x:[0,T]\to\mathbb{R}$, and $p\ge 1$, we define the $p$-variation of $x$ by $$\|x\|_p := \sup_{\pi}\left(\sum_{[s,t]\in\pi} |x_t - x_s|^p\right)^{1/p},$$ where the supremum is taken over all (finite) partitions of $[0,T]$. (Note that…
Analysis801
- 589
1
vote
1 answer
An exercise about p-variation
I have a question about p-variation.
I'm reading A first Course in Sobolev Spaces by Giovanni Leoni. In this book, a confused construction is demanded (Exercise 2.26):
Let $p\ge1$.Given a function $u:[a,b]\rightarrow\mathbb{R}$ for every…
1
vote
2 answers
Topology on space of full signatures in rough path theory
In Theorem 3.1 of this paper, the following result is formulated:
Suppose $f:S_1 \to \mathbb{R}$ is a continuous function where $S_1$ is a compact subset of $S(\mathcal{V}^p(J,E))$. Then for any $\epsilon > 0$, there exists a linear functional $L…
zhy
- 13
1
vote
0 answers
Two-way anova or One-way anova
I have to tested multiple reinforcement learning systems and record their performances [test-accuracy] based on two categories of [system-type] and [communication-level]. The system-type can be either one of the two types namely "centralized" or…
Hirad Gorgoroth
- 111
1
vote
1 answer
How to show the 2-variation of Brownian motion sample paths is infinite
Brownian motion has bounded quadratic variation, however for almost every sample path, the $p$-variation is infinite for any $p>1/2$, where the $p$ variation takes the supremum over all possible partitions for a given path.
How can I prove this is…
Daven
- 974
0
votes
0 answers
p-Variation of a Semimartingale
I have two questions regarding the $p$-Variation of a Semimartingale:
Let $X_t$ be a semimartingale on $[0,1]$ and $\Pi_n = \{t^n _k = \frac{k}{n}: 0 \leq k \leq n\} $ a partition of $[0,1]$.
For $p >0$ we define the $p$-Variation-sum $V_{n,p}$…
K123
- 31
0
votes
1 answer
Question: For one survey (Combinatorics)
For one survey 39 students from first, second and third year must be chosen, so that at least 8 students are from first year, and at least 3 students from second year. How many possible ways are there to make this? I tried to use the combinations…
Pol
- 21