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Questions tagged [rough-paths]

Questions about an area of probability theory, rough paths.

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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 Ito process $dX_t = \...
Bombadil's user avatar
4 votes
1 answer
178 views

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_+} \...
Nate River's user avatar
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2 votes
1 answer
58 views

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|^\...
user479223's user avatar
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1 vote
1 answer
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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)) :...
Gaspar's user avatar
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25 views

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 $\Delta x$...
Gaspar's user avatar
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3 votes
2 answers
122 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([...
Gaspar's user avatar
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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 ...
Theo Diamantakis's user avatar
4 votes
1 answer
201 views

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 ...
NZK's user avatar
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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\...
tsnao's user avatar
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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-...
user479223's user avatar
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2 votes
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92 views

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 ...
user479223's user avatar
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5 votes
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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 ...
Theo Diamantakis's user avatar
1 vote
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106 views

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)...
user479223's user avatar
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158 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 ...
PhD_InStochastics's user avatar
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1 answer
156 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 ...
user479223's user avatar
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2 votes
0 answers
116 views

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 ...
anatolvitold's user avatar
3 votes
1 answer
425 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( \...
anatolvitold's user avatar
3 votes
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124 views

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{...
Martin Geller's user avatar
2 votes
0 answers
212 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}^...
almosteverywhere's user avatar
5 votes
1 answer
182 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}), \...
Bernard_Karkanidis's user avatar
6 votes
0 answers
335 views

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 ...
John Rached's user avatar
1 vote
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159 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 $H>1/2$, ...
yassine yassine's user avatar
5 votes
1 answer
554 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 ...
Creator's user avatar
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9 votes
1 answer
608 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 ...
Oleg's user avatar
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5 votes
2 answers
631 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 ...
Furdzik Zbignew's user avatar
0 votes
1 answer
142 views

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 $\...
ABIM's user avatar
  • 4,775
1 vote
1 answer
176 views

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 (...
Roman22's user avatar
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2 votes
0 answers
52 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?
ABIM's user avatar
  • 4,775
3 votes
0 answers
68 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 ...
ABIM's user avatar
  • 4,775
2 votes
1 answer
402 views

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 ...
Abani Sarma's user avatar
7 votes
0 answers
319 views

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 ...
user avatar
3 votes
0 answers
85 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 ...
user avatar
3 votes
1 answer
214 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 ...
Creator's user avatar
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5 votes
1 answer
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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 ...
Creator's user avatar
  • 435
7 votes
1 answer
898 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. ...
user avatar
3 votes
1 answer
289 views

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=...
Guohuan Zhao's user avatar
2 votes
0 answers
108 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 $\...
quallenjäger's user avatar
3 votes
1 answer
309 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?
ABIM's user avatar
  • 4,775
5 votes
0 answers
231 views

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 $M$...
Matthias Ludewig's user avatar
2 votes
0 answers
151 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_{|\...
Matthias Ludewig's user avatar
4 votes
3 answers
751 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 ...
user avatar
3 votes
2 answers
466 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 ...
user avatar
8 votes
2 answers
1k 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}...
user avatar
25 votes
2 answers
3k 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 ...
kenneth's user avatar
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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 ...
Jedi's user avatar
  • 203
9 votes
1 answer
2k views

An iterated tensor product integral

In "Differential equations driven by rough paths" (Terry Lyons, et al) section 1.4.2 it's claimed that the symmetric part of the tensor: $\int_{0 \le u_1 \le \cdots \le u_j \le t} \mathrm{d}X_{u_1} \...
Pablo Lessa's user avatar
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