# Questions tagged [rough-paths]

Questions about an area of probability theory, rough paths.

46
questions

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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 = \...

4
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1
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178
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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_+} \...

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1
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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|^\...

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1
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87
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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)) :...

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0
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25
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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 $\Delta x$...

3
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2
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### 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([...

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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 ...

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201
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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 ...

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0
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92
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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\...

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0
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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-...

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92
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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 ...

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116
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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 ...

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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)...

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158
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### 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 ...

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1
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156
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### 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 ...

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0
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116
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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 ...

3
votes

1
answer

425
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### 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( \...

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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{...

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### 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}^...

5
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182
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### 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}), \...

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335
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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 ...

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159
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### 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
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1
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554
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### 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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1
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608
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### 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 ...

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2
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631
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### 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 ...

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1
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142
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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
$\...

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1
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176
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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 (...

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0
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### 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?

3
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### 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 ...

2
votes

1
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402
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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 ...

7
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0
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319
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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 ...

3
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0
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85
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### 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 ...

3
votes

1
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214
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### 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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1
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459
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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 ...

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1
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898
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### 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. ...

3
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1
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289
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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=...

2
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0
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108
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### 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 $\...

3
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1
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309
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### 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
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0
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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 $M$...

2
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0
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151
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### 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_{|\...

4
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3
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751
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### 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 ...

3
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2
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466
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### 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 ...

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2
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### 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}...

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2
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3k
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### 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 ...

10
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1
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1k
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### 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 ...

9
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1
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2k
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### 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} \...