Questions tagged [rough-paths]
Questions about an area of probability theory, rough paths.
51 questions
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+50
Local Lipschitz continuity of signature map $S:C^{1-\text{var}}([0,T],\mathbb{R}^d) \to \mathcal{H}$
Just came across the claim that the signature map (between path space and tensor space) is locally Lipschitz continuous with respect to the $1-$variation norm (see section A.2.1).
More specifically, ...
- 161
3
votes
0
answers
87
views
Functorial relationships between Hopf algebras and rough paths
In rough paths theory, the signature of a path $x: [0, T] \to \mathbb{R}^d$ is an element in the tensor algebra $T((\mathbb{R}^d))$. These signatures reside within the group-like elements and ...
- 232
8
votes
1
answer
449
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What do smooth signatures give you?
My background is in rough paths theory.
In short, if you have an irregular function $f:[0,T]\to\mathbb R^d$ and you want to make sense of integrals $\int_s^t \cdot \ df(r)$, the right objects that are ...
- 1,904
3
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0
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77
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Is the norm of first or second level of of signature a convex function?
I understand this is not a research level question but I really want to know, would anyone please help.
This question is related to the signatures that arises in rough path theory. https://en....
- 495
3
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1
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248
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Can any function in $C^\alpha$ be approximated in $C^{\alpha^-}$ by singular functions?
For every positive $\alpha < 1$, we consider the space $C^{\alpha}$ of Holder continuous functions of order $\alpha$ on $[0, 1]$, equipped with the norm
$$\|f\|_{C^\alpha} := \sup|f| + \sup_{x, y \...
- 6,155
2
votes
1
answer
311
views
Conditional expectation w.r.t. filtration of Brownian motion as a continuous map of its paths
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space on which we define Brownian motion $B$ and let us denote by $\mathcal{F}_t$ its natural filtration. Assume we have Itô process $dX_t = \...
- 21
4
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1
answer
201
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How much can you improve a Hölder function by composing it with another?
Let $f: [0, 1] \to \mathbb R$ be a continuous function. Define the local Hölder exponent $H(f, x)$ of $f$ at $x \in [0, 1]$ by
$$H(f, x) := \sup\left\{0 \leq \alpha \leq 1\mid\lim_{\delta \to 0_+} \...
- 6,155
3
votes
1
answer
79
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Can a lift satisfy Chen's relation, geometric condition but not be a rough path?
Let $(X,\mathbb X):[0,1]^2\to \mathbb R^d\oplus\mathbb R^{d\times d}$ satisfy the following four properties:
\begin{align}
&X_{s,t}=X_{0,t}-X_{0,s}\\
&\sup_{t\neq s}\frac{|X_{s,t}|}{|t-s|^\...
- 1,904
1
vote
1
answer
139
views
Carnot–Carathéodory norm and the inner product norm
It is well-known that given the extended tensor algebra $T((\mathbb{R}^d))$ one may extract a separable Hilbert space by considering the subset
$$T^1((\mathbb{R}^d)) := \left\{h \in T((\mathbb{R}^d)) :...
- 161
1
vote
0
answers
37
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Inner product of signatures of piecewise linear paths
It is a well-know observation that, given two points $x_1,x_2 \in \mathbb{R}^d$, the path signature associated to their linear interpolation is given by the tensor exponential. Precisely, if $\Delta x$...
- 161
3
votes
2
answers
144
views
Estimate on $\alpha$-Hölder norm of path signature
Let $N \geq \lfloor 1/\alpha \rfloor > 0$ and consider a weakly geometric $\alpha$-Hölder rough path $\textbf{x}$ that preserves the origin, i.e. an element $\textbf{x} \in C^{\alpha\text{-Höl}}_o([...
- 161
3
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0
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89
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Cylindrical Wiener processes or SPDE that can make use of Banach valued rough paths?
Rough paths theory has an often advertised perk that it mostly works for general Banach spaces. I am trying to think of some nice examples that actually use this feature, and am coming up stuck.
The ...
- 193
4
votes
1
answer
235
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Truncated fixed point and regularity structures
This question arose via the helpful comments on this earlier question.
In Hairer's theory of regularity structures, fixed point problems are first solved in certain spaces $D^\gamma$ which consist of ...
- 447
0
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0
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101
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Integration with respect to $B_H(t) B_H(s) - \mathbb{E} \{ B_H ( t ) \, B_H ( s) \}$
The time-derivative $\frac{dB_H}{dt}$ of the fractional Brownian motion may be interpreted as a random Schwartz distribution acting on a test function by
$$
\left\langle \frac{dB_H}{dt}, f \right\...
- 620
2
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0
answers
58
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Does uniform convergence on compacts of drifts in rough differential equation imply convergence of solutions?
Consider the RDE
$$dY^n=b_n(Y^n) \, dt+\sigma(Y^n) \, d\mathbf X$$
where $\mathbf X$ is a rough path, $\sigma$ is as smooth as you'd like and $b_n$ are Lipschitz. If $b_n\to b$ uniformly then Friz-...
- 1,904
3
votes
0
answers
101
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What is the state of the art for rough path regularity on coefficients?
Consider the rough differential equation
$$dY_t=b(Y_t,t) \, dt+\sigma(Y_t,t) \, d\mathbf X_t,$$
where $\mathbf X$ is a $p$-rough path with $1\leq p<3$. If $b$ and $\sigma$ are $C^3_b$ then we have ...
- 1,904
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0
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146
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What do the Carnot groups act on?
My question is in some sense a less ambitious version of the following MO question where the answer was inconclusive.
A Carnot group of step $N$ can be identified within the tensor algebra, modulo ...
- 193
2
votes
1
answer
153
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Interpolation theorem for general rough paths
In Friz and Hairer's notes on rough paths, there is exercise 2.9 which is called the "interpolation theorem". It says that if you have a sequence of rough paths $\mathbf X^n=(X^n,\mathbb X^n)...
- 1,904
0
votes
1
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163
views
Stability of SDE fBM
Consider an n-dimensional Ito process
$$
X_t^x = x + \int_0^t\, \alpha(s)ds + \int_0^t\,\beta(s)\,dB^H(s),
$$
where $1/3<H<1$ is the Hurst parameter for an $n$-dimensional fractional Brownian ...
0
votes
1
answer
161
views
Let $X^n$ be a collection of smooth functions so that their $\alpha$-Holder norms are uniformly bounded
Let $X^n$ be a collection of smooth functions so that their $\alpha$-Holder norms for $\alpha \in (1/3,1/2)$ are uniformly bounded - that is $\sup_n \|X^n\|_\alpha<\infty$. Define the standard ...
- 1,904
2
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0
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124
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Rough path expected signature vs cumulant-generating function / characteristic function
What is the point of using rough path expected signature to characterize the law of а stochastic process when the cumulant generating function is known ($\log\mathbb{E}[e^{i\theta X(t)}]$)?
Since an ...
- 53
3
votes
1
answer
531
views
What is a tensor product of random variables?
I am trying to understand the the following paper https://arxiv.org/pdf/1810.10971.pdf, in particular Example 2:
If $ Y \sim N(0,1)$, the standard normal on $\mathbb{R}$, then
$ \begin{align*} \Big( \...
- 53
3
votes
0
answers
126
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A path with zero increments and positive area
I am studying rough paths from the 2007 St Flour lecture notes and I came across the example at the end of chapter one of the sequence of paths $X(n):[0,2\pi]\to \mathbb R^2$ given by $X_t(n) = \frac{...
- 177
3
votes
1
answer
297
views
Choice of stochastic integral picking the forward point in Riemann sum approximation and reversibility?
Consider the standard Riemann sum approximation of a stochastic integral (w.r.t Brownian motion for example) which is given by
\begin{align}
\int_0^t \sigma(X_s) \circ_{\lambda}dB_s \approx \sum_{i=1}^...
- 175
5
votes
1
answer
187
views
Regularity of law of conditional law of a Markov process equivalent to regularity of its paths
Let $(X_t^x)_{t\in [0,\infty),\,x\in \mathbb{R}^n}$ be a Markov process taking values in $\mathbb{R}^m$ and defined on some stochastic basis $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in [0,\infty}), \...
8
votes
0
answers
353
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Rough paths, unparametrized path space, and Kontsevich's moduli space of stable maps
Let $X$ be a manifold. Modulo reparametrization, the path space of $X$ is a groupoid $\Pi_X$. In Kapranov's "Free Lie Algebroids and the Space of Paths", Kapranov constructs an associated ...
- 31
1
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0
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166
views
Are SDE adapted to the natural filtration?
Let $(B^H_t)_{t\in [0,T]}$ be a fractional Brownian motion. We consider the following SDE where $b$ and $\sigma$ are Lipschitz
$$X_t=x+\int_0^t b(X_s)ds+\int_0^t\sigma(X_s)dB^H_s.$$
When $H>1/2$, ...
5
votes
1
answer
589
views
How to compare pathwise convergence and convergence in probability
This question was asked quite sometime back in mathexchange and deleted, as it was downvoted, asked again but never got an answer. So I am asking here.
Motivation: It appears pathwise convergence can ...
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9
votes
1
answer
621
views
Uniqueness of solutions of Young differential equations
Consider the following one dimensional Young differential equation:
\begin{align*}
&Y_t=\int_0^t Y_s dX_s,\quad t\in[0,1];\\
&Y_0=0.
\end{align*}
Here the driving process $X$ is a bounded ...
- 931
5
votes
2
answers
697
views
Intuition behind Gubinelli derivative
I apologise for the confusion of the following sentences. I'm lazy to give more information about Rough path theory as Is a fairly broad subject.
On page 14 of "A Course on Rough Paths
With an ...
- 232
0
votes
1
answer
157
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Signature Map From $p$-Geometric Rough Paths to $T(\mathbb{R})$
Let $f:[0,T]\rightarrow \mathbb{R}^d$ be a p-geometric rough path and let $\mathcal{G}_p^d$ be the collection of all such paths. Does the Lyons signature map define a continuous bijection between
$\...
- 5,405
1
vote
1
answer
205
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Rough paths theory for Non-Markovian processes
I would like to know whether there is a suitable extension of the theory of rough paths that could be useful to solve Non-Markovian problems.
I would appreciate any example or also any other theory (...
- 13
2
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0
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56
views
Is the space of $p$-geometric rough paths is Homeomorphic to Frechet Space
Let $\Omega G^p([0,T];\mathbb{R}^n)$ be a space of $p$-geometric rough paths with values in $\mathbb{R}^n$. Is $\Omega G^p([0,T];\mathbb{R}^n)$ homeomorphic to some Fr\'{e}chet space?
- 5,405
3
votes
0
answers
75
views
p-Variation distance defines semi-martingales
Question
When, does the process $\tilde{X}_t$, defined path-wise by
$$
\tilde{X}_t(\omega)\triangleq \rho_{\frac1{2}}\left((y_t,\mathbb{Y}_t),(x_t(\omega),\mathbb{X}_t(\omega))\right),
$$
define a ...
- 5,405
2
votes
1
answer
440
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Can we extract information from signature (rough path theory) to construct part of signal?
This question is related to rough path theory. Consider we have obtained signature obtained from a set discrete data points postulating linear from one data point to another. Such signature are used ...
- 21
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0
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334
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What are morphisms between regularity structures?
In Hairer's notes A Theory of Regularity Structures he defines automorphisms of a regularity structure on page 28. I will recall the definition here:
Is there any way of extending this to morphisms ...
user69208
3
votes
0
answers
89
views
Why is the Jain Monrad condition the right condition on general Gaussian processes?
Consider a covariance function $\sigma^2(s,t)=E((X_t-X_s)^2)$, where $X\colon I\to \Bbb R^d$ is a Gaussian process.
Given a $\rho\ge 1$ and a superadditive function $\omega(s,t)$ we say that Jain ...
user69208
3
votes
1
answer
228
views
Are Holder Condition and signal to noise ratio (SNR) related?
This question was posted in https://math.stackexchange.com but I got hardly any view. If posting here is an objection please let me know I would delete it immediately.
This question has evolved from ...
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5
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1
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544
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Under what condition we get back path from signatures in rough path theory?
A link to wikipedia for rough pat theory is: https://en.wikipedia.org/wiki/Rough_path
It appears path and signatures has one to one mapping in many cases. I understand that the signature is not ...
- 495
7
votes
1
answer
972
views
What does the group action of a rough path in a Lie group look like?
Rough paths can be thought of as taking values in a Lie group embedded in the tensor algebra of $\Bbb R^d$. See page 17/section 2.3. Lie groups represent the continuous symmetries of some object. ...
user69208
3
votes
1
answer
311
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An integral by rough path.
If $(b, \mathbb{b})\in \mathcal{D}^{\alpha}[0,T],\ \alpha\in (\frac{1}{3}, \frac{1}{2})$. $\mathcal{D}^{\alpha}[0,T]$ is the space of those rough paths $(b,\mathbb{b})$
such that
$$ \|b\|_\alpha=...
- 515
2
votes
0
answers
111
views
Iterated integral with a irregular path
For the proof of Fundamental Lemma 3.1 on the page 400 of K.T. Chen's 1957 paper Integration of paths--A faithful representation of paths by noncommutative
formal power series, it requires the path $\...
- 101
3
votes
1
answer
346
views
Reference: Ito lemma for rough paths
Hi I'm looking for an Ito-type lemma for rough paths but am having difficulty finding something. Could someone kindly point me in the right direction?
- 5,405
6
votes
0
answers
245
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Second order calculus and rough paths
In Emery's book "Stochastic calculus in manifolds", he shows how to make sense of integrals of the form
$$ \int \langle\Theta_t, \mathbf{d} X_t\rangle,$$
where $X$ is a semimartingale on a manifold $M$...
- 9,853
2
votes
0
answers
161
views
Continuity of solution map to Stratonovich Integral
For paths $x:[0, T] \rightarrow \mathbb{R}^n$, the Stratonovich integral along a one form $\omega$ on $\mathbb{R}^n$ can be defined by
$$ S_\omega(x) := \int_0^T \omega(x(t)) \mathrm{d}x(t) := \lim_{|\...
- 9,853
4
votes
3
answers
796
views
What's an example of a rough path that's not Ito/Stratonovich-Brownian rough path?
The only rough path that I've ever seen discussed are the ones associated with Brownian motion. I could use a "rough path" for any nice function, defeating the point. In particular are there ...
user69208
3
votes
2
answers
477
views
Rough path theory- Verify that $\mathbb{X}_{s,t}=\int_s^t X_{s,r} \otimes dX_r$
This is exercise 7.7 from Martin Hairer's Rough Path notes.
Verify that $\mathbb{X}_{s,t}=\int_s^t X_{s,r} \otimes dX_r$ where the integral is to be interpreted in the sense of (4.22) (I'll define ...
user69208
8
votes
2
answers
2k
views
Why the term "geometric" rough path?
A "geometric" rough path is a rough path such that $Sym(\mathbb{X}_{s,t})=\frac{1}{2}X_{s,t}\otimes X_{s,t}$. For example the Ito rough path is not geometric because $Sym(\mathbb{X}_{s,t})=\frac{1}{2}...
user69208
25
votes
2
answers
4k
views
Understanding of rough path
A rough path is defined as an ordered pair
$ (X, \mathbb X)$, where $X$ is a path mapping from $[0,T]$ to some Banach space $V$
and $\mathbb X:[0,T]^2 \mapsto V^2$ is another mapping for additional ...
- 1,399
10
votes
1
answer
1k
views
Understand rough path iterated integral and how to compute it numerically?
The "signature" of rough path theory is defined by iterated integral as
$s(k)=\int_{0 \le u_1 \le \cdots \le u_k \le t} \mathrm{d}X_{u_1} \otimes \cdots \otimes \mathrm{d}X_{u_k}$
in witch $X(t)$ is ...
- 203