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Questions tagged [rough-paths]

Questions about an area of probability theory, rough paths.

2
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0answers
41 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 ...
1
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0answers
27 views

Can we relate Levy area to the resonant term in paracontrolled calculus in the case of a controlled differential equation?

Consider the SDE: $$dX_t=V(X_t)dW_t^H$$ Where $W_t^H$ is fractional Brownian motion with Hurst parameter $H\in (1/3,1/2)$. This equation can be solved by rough paths theory. The idea is we have to ...
4
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0answers
78 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 ...
2
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0answers
60 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 ...
3
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1answer
141 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 ...
3
votes
1answer
88 views

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 ...
2
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1answer
244 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. ...
2
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0answers
116 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=...
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0answers
67 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 $\...
3
votes
1answer
136 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
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0answers
133 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 $M$...
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0answers
77 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_{|\...
3
votes
3answers
408 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 ...
1
vote
2answers
255 views

Rough path theory- Verify that $\mathbb{X}_{s,t}=\int_s^t X_{s,t} \otimes dX_r$

This is exercise 7.7 from Martin Hairer's Rough Path notes. Verify that $\mathbb{X}_{s,t}=\int_s^t X_{s,t} \otimes dX_r$ where the integral is to be interpreted in the sense of (4.22) (I'll define ...
6
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1answer
466 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}...
14
votes
1answer
1k 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 ...
7
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1answer
687 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 ...
5
votes
1answer
901 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} \...