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# Questions tagged [rough-paths]

Questions about an area of probability theory, rough paths.

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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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### 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 $H>1/2$, ...
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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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### 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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### 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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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 $\... 1 vote 1 answer 141 views ### Rough paths theory for Non-Markovian processes I would like to know whether there is a suitable extension of the theory of rough paths that could be useful to solve Non-Markovian problems. I would appreciate any example or also any other theory (... 2 votes 0 answers 50 views ### Is the space of$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)$homeomorphic to some Fr\'{e}chet space? 2 votes 0 answers 63 views ### p-Variation distance defines semi-martingales Question When, does the process$\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 ... 2 votes 1 answer 316 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 ... 7 votes 0 answers 282 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 ... 3 votes 0 answers 83 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 votes 1 answer 188 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 ... 5 votes 1 answer 356 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 ... 7 votes 1 answer 601 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. ... 3 votes 1 answer 259 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 $$\|b\|_\alpha=... 2 votes 0 answers 103 views ### Iterated integral with a irregular path For the proof of Fundamental Lemma 3.1 on the page 400 of K.T. Chen's 1957 paper Integration of paths--A faithful representation of paths by noncommutative formal power series, it requires the path \... 3 votes 1 answer 277 views ### Reference: Ito lemma for rough paths Hi I'm looking for an Ito-type lemma for rough paths but am having difficulty finding something. Could someone kindly point me in the right direction? 5 votes 0 answers 206 views ### Second order calculus and rough paths In Emery's book "Stochastic calculus in manifolds", he shows how to make sense of integrals of the form$$ \int \langle\Theta_t, \mathbf{d} X_t\rangle,$$where X is a semimartingale on a manifold M... 2 votes 0 answers 138 views ### Continuity of solution map to Stratonovich Integral For paths x:[0, T] \rightarrow \mathbb{R}^n, the Stratonovich integral along a one form \omega on \mathbb{R}^n can be defined by$$ S_\omega(x) := \int_0^T \omega(x(t)) \mathrm{d}x(t) := \lim_{|\... 4 votes 3 answers 663 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 ... 3 votes 2 answers 441 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 ... 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}... 2k 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 ...
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 ...