Tagging - 2018-05-19

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12:39 PM

A new tag rough-path-theory.

In stochastic analysis, a rough path is a generalization of the notion of smooth path allowing to construct a robust solution theory for controlled differential equations driven by classically irregular signals, for example a Wiener process. The theory was developed in the 1990s by Terry Lyons. Several accounts of the theory are available. Rough path theory is focused on capturing and making precise the interactions between highly oscillatory and non-linear systems. It builds upon the harmonic analysis of L.C. Young, the geometric algebra of K.T. Chen, the Lipschitz function theory of H. Whitney...

Q: Could someone explain rough path theory? More specifically, what is the higher ordered "area process" and what information is it giving us?

http://www.hairer.org/notes/RoughPaths.pdf here is a textbook, but I am completely lost at the definition. It is defined on page 13, chapter 2. A rough path is defined as an ordered pair, $(X,\mathbb{X})$ where $X$ is a continuous process and $\mathbb{X}$ is a higher ordered area process which de...

Q: How to calculate signatures numerically in rough path theory.

The question is related to the link:https://en.wikipedia.org/wiki/Rough_path#Signature I search the internet looking for tutorial on how to calculate the signature sequence, but could not find a clear document. The first term is one the second term seems to be $X_t−X_s$ but after the 2nd term ho...

reference-request soft-question sde rough-path-theory

Q: Is there a relation between derivative of a smooth path and the signature?

This question is related to rough path theory. A link to wikipedia is: https://en.wikipedia.org/wiki/Rough_path As I understand a signature determines a path. For this question consider a one dimensional smooth path and a corresponding signature. The question is: can there be relation between th...

stochastic-processes banach-spaces stochastic-analysis sde rough-path-theory

Q: Hölder-type estimate for two parameter function

I am studying the book 'A course on Rough Paths' by Friz and Hairer at the moment and I have a problem with one conclusion they're making on page 11 (also described below). Their claim is the following: Let $X:[0,T]^2\rightarrow V$, $T>0$, $V$ Banach space satisfy the following: $X(s,t) \leq C...