All Questions
Tagged with stochastic-calculus rough-paths
12 questions
3
votes
0
answers
77
views
Is the norm of first or second level of of signature a convex function?
I understand this is not a research level question but I really want to know, would anyone please help.
This question is related to the signatures that arises in rough path theory. https://en....
- 495
2
votes
1
answer
311
views
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 Itô process $dX_t = \...
- 21
3
votes
1
answer
79
views
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|^\...
- 1,904
0
votes
0
answers
101
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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
$$
\left\langle \frac{dB_H}{dt}, f \right\...
- 620
0
votes
1
answer
163
views
Stability of SDE fBM
Consider an n-dimensional Ito process
$$
X_t^x = x + \int_0^t\, \alpha(s)ds + \int_0^t\,\beta(s)\,dB^H(s),
$$
where $1/3<H<1$ is the Hurst parameter for an $n$-dimensional fractional Brownian ...
3
votes
0
answers
126
views
A path with zero increments and positive area
I am studying rough paths from the 2007 St Flour lecture notes and I came across the example at the end of chapter one of the sequence of paths $X(n):[0,2\pi]\to \mathbb R^2$ given by $X_t(n) = \frac{...
- 177
3
votes
1
answer
297
views
Choice of stochastic integral picking the forward point in Riemann sum approximation and reversibility?
Consider the standard Riemann sum approximation of a stochastic integral (w.r.t Brownian motion for example) which is given by
\begin{align}
\int_0^t \sigma(X_s) \circ_{\lambda}dB_s \approx \sum_{i=1}^...
- 175
3
votes
0
answers
75
views
p-Variation distance defines semi-martingales
Question
When, does the process $\tilde{X}_t$, defined path-wise by
$$
\tilde{X}_t(\omega)\triangleq \rho_{\frac1{2}}\left((y_t,\mathbb{Y}_t),(x_t(\omega),\mathbb{X}_t(\omega))\right),
$$
define a ...
- 5,405
3
votes
0
answers
89
views
Why is the Jain Monrad condition the right condition on general Gaussian processes?
Consider a covariance function $\sigma^2(s,t)=E((X_t-X_s)^2)$, where $X\colon I\to \Bbb R^d$ is a Gaussian process.
Given a $\rho\ge 1$ and a superadditive function $\omega(s,t)$ we say that Jain ...
user69208
3
votes
1
answer
311
views
An integral by rough path.
If $(b, \mathbb{b})\in \mathcal{D}^{\alpha}[0,T],\ \alpha\in (\frac{1}{3}, \frac{1}{2})$. $\mathcal{D}^{\alpha}[0,T]$ is the space of those rough paths $(b,\mathbb{b})$
such that
$$ \|b\|_\alpha=...
- 515
3
votes
1
answer
346
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?
- 5,405
6
votes
0
answers
245
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 $M$...
- 9,853