13
votes

Accepted

### What does the group action of a rough path in a Lie group look like?

Though I like the Arnoldian spirit of the question ("a group is not some set with a forgettable system of axioms but something which acts on a space"), I think the comments given above are already ...

6
votes

Accepted

### Truncated fixed point and regularity structures

Before I try to answer the broader question let me try to clear up a potential confusion. The truncations that appear at the level of modelled distributions don't really change the distribution they ...

6
votes

Accepted

### Intuition behind Gubinelli derivative

In a way it is very much like a usual derivative. Recall first that for a regular function $Y$, its derivative $Y'_s$ at a point $s$ is the (unique) number such that
$$
Y_{t,s}=Y'_s(t-s)+ R_{s,t},
$$
...

6
votes

Accepted

### Under what condition we get back path from signatures in rough path theory?

Loops don't get canceled out in the signature. (You might like to compute the signature of a circle. The second iterated integral is nonzero; in fact, by Green's theorem, it gives you the area inside ...

5
votes

### Why the term "geometric" rough path?

I agree with Martin's answer - but there were other additional and compelling reasons.
A rough path (as in the original papers) was a path in the tensor algebra over V with appropriate "p-...

5
votes

### Understanding of rough path

You have seen all kinds of integration theories before — Itô, Stratonovich, and I'm sure plenty others. Rough paths takes a step back and asks what we want from an integration theory. And so long as ...

5
votes

Accepted

### How much can you improve a Hölder function by composing it with another?

For any $\alpha>0$, put $$f(x)=\begin{cases}-e^{-\frac{1}{x^2}},&x<0\\x^\alpha,&x>0\end{cases}\quad \text{and}\quad F(x)=\begin{cases}x,&x<0\\e^{-\frac{1}{x^2}},&x>0\end{...

4
votes

Accepted

### Can any function in $C^\alpha$ be approximated in $C^{\alpha^-}$ by singular functions?

Here is a concrete approximation. Note that there are variants of the Cantor function that are Hölder continuous with exponents arbitrarily close to $1$. Specifically, for $0\le\sigma<1$, consider ...

4
votes

### Intuition behind Gubinelli derivative

We want to define $\int_0^T f(X_s) dX_s$ for smooth bounded $f$ with bounded derivatives of all orders. Using linearity and a partition ${t_k}$ of $[0,T]$, we have
\begin{align*}\int_0^T f(X_s) dX_s&...

4
votes

### Can we extract information from signature (rough path theory) to construct part of signal?

Inverting the signature is an ongoing subject of research. The basic version of your final question is "If I have a d-dimensional path starting at the origin which is N points joined by linear ...

4
votes

Accepted

### Are Holder Condition and signal to noise ratio (SNR) related?

The Hölder exponent $h$ and the Hurst exponent $H$ (related by $h=H-1$ for a continuous signal) characterise correlations in the noise, not the size of the noise (as quantified by the signal-to-noise ...

3
votes

Accepted

### Rough paths theory for Non-Markovian processes

It really depends on what sort of non-Markovian equations you have in mind, but it does certainly allow you to give solution theories for SDEs driven by fractional Brownian motion with Hurst parameter ...

2
votes

Accepted

### Let $X^n$ be a collection of smooth functions so that their $\alpha$-Holder norms are uniformly bounded

This cannot hold, and in a sense rough path theory has to be developed precisely because of this reason; otherwise, rough path lifts would be defined uniquely for any curve of Hölder regularity $>1/...

2
votes

Accepted

### What is a tensor product of random variables?

In this context, I believe the tensor product on random variables is nothing other than the tensor product over the values of the RVs. (In other words, if $\Omega$ is a sample space and $X : \Omega \...

2
votes

Accepted

### Signature Map From $p$-Geometric Rough Paths to $T(\mathbb{R})$

The signature is continuous on the space of $p$-geometric rough paths, but it is not injective since it is parametrisation-independent and invariant under concatenation with "tree-like" pieces. ...

2
votes

### Estimate on $\alpha$-Hölder norm of path signature

Yes, we see this by following their definition of $C$ in the proofs.
In lemma 9.2, they bound in (9.1) by $\frac{2^{N}}{(N+1)!}\ell^{N+1}=B_{N}\ell^{N+1}$. This $B_{N}\to 0$ as $N\to +\infty$. Then in ...

2
votes

Accepted

### Can a lift satisfy Chen's relation, geometric condition but not be a rough path?

Indeed, this can fail. Take any geometric rough path lift $\mathbb X^{ij}$ and simply add in a non-symmetric way the increment of a function $F$ that fails to be $2\gamma$-Hölder. For example, take ...

1
vote

Accepted

### Conditional expectation w.r.t filtration of Brownian motion as a continuous map of its paths

The answer to the question as stated is no. Take the SDE coefficients $\alpha$ and $\beta$ to be deterministic, then $X_t$ is $\mathcal F_t$ measurable, so that $\mathbb E[X_t | \mathcal F_t] = X_t$ ...

1
vote

### Carnot–Carathéodory norm and the inner product norm

I think the bound you are looking for cannot hold because of scaling.
Here's an attempt at an argument: fix a path $\textbf{x}\in C^\alpha$ (for simplicity I assume $\alpha\in(1/2,1)$), and take $\...

1
vote

### Stability of SDE fBM

As explained in the comments the Lip-continuity just follows from the linearity.
Similarly, for Gaussian, we can use that $\beta\in C^{1}$ and we can define the Young integral which is an ...

1
vote

Accepted

### Regularity of law of conditional law of a Markov process equivalent to regularity of its paths

No, this has no reason to be true. Take for example $X_t^x = x+t$ for $x \ge 0$ and $x-t$ for $x < 0$ ($n=1$). Paths are smooth, but $f$ is discontinuous at $x=0$.

1
vote

### How to compare pathwise convergence and convergence in probability

As I did not get any reply, I am trying to verify my understanding. This is related to convergence of SDE/PSDE through Ito type calculous and rough path or regularity structure. Any comments would be ...

1
vote

### Uniqueness of solutions of Young differential equations

We will follow the Lemma 8.10. (Rough Gronwall) and Proposition 8.12 from "a course in rough paths" but modify them for this particular setting of studying
$$Y_{t}=Y_{0}+\int^{t}_{0} Y dX,$$
...

1
vote

### An integral by rough path.

This is an old question but it's quite interesting. Here is an attempt with some extra assumptions because $f$ and $b$ are not concrete.
The reasonable approach is to try to find a Gubinelli ...

1
vote

Accepted

### Reference: Ito lemma for rough paths

In addition to what Nate Eldredge mentioned, here is a recent paper that you might like to consider.
Keller, Christian; Zhang, Jianfeng Pathwise Itô calculus for rough paths and rough PDEs with path ...

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