20
votes
Accepted
Is there any reason to use paracontrolled calculus over regularity structures?
I don't think that the reason given in the paper by Bailleul and Bernicot is a good one. Basically, they treat an example which is simple enough so that it is still manageable to describe the various ...
- 10.3k
18
votes
What phenomena are better modelled by SDE instead of ODE?
This is a really broad question, but in general noise terms become important if there are few degrees of freedom; for example, chemical reaction kinetics can be accurately described by coupled ODE's, ...
- 188k
15
votes
Why do stochastic integrals depend on the choice of partitioning points?
First, note that the right comparison is not with the Riemann integral but rather with the Riemann-Stieltjes integral.
To be concrete, consider $\int_0^1 X_s dW_s$ where $W$ is Brownian motion
and $...
- 7,514
15
votes
Accepted
A curious martingale
No. see Martingale Convergence
Theorem 4 Let X be a continuous martingale. Then, almost surely, one of the following is satisfied
$X_\infty=\lim_{t\rightarrow\infty}X_t$ exists and is finite.
$\...
- 5,474
11
votes
White noise vs. black noise
Percolation 'noise' is generated by a perfectly good family of random variables, the 'quad-crossing' events. Basically, for every diffeomorphic image of the unit square, this random variable is $1$ if ...
- 10.3k
11
votes
What phenomena are better modelled by SDE instead of ODE?
Brownian motion is an obvious example. Brownian motion described particles dispersed in a liquid that are large enough that the random jossling of the water molecules becomes important. Being one of ...
9
votes
Accepted
Kolmogorov continuity theorem and Holder norm
One can apply a deterministic result, called Garsia--Rodemich--Rumsey inequality, to estimate $\mathrm{E}[||X||^\alpha_{\gamma;[0,T]}]$. Here is a particular form of this result, which is most ...
- 1,533
9
votes
Accepted
Why do we mainly integrate with respect to martingales?
I am aware that this is a really old post, but I feel it is a very important question. My personal favourite answer to this question lies in the theory of rough paths. I will give both a technical ...
- 6,155
9
votes
Accepted
Maxima of Brownian motion
A good way to measure the set of maxima is the Hausdorff dimension of the set of records, which for BM is a.s. 1/2. Because of time/scale invariance, the dimension is the same for $\alpha B_\cdot$, ...
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9
votes
Accepted
Intuition/elegant reason for why Langevin diffusion converges to $\exp(-U)$?
The reason a Langevin diffusion leaves $\nu(x)=e^{-U(x)}$ invariant is because it is symmetric or reversible with respect to $\nu$. In comparison to general diffusion processes, the ergodic ...
- 6,045
9
votes
Accepted
Definition of infinite-dimensional Gaussian random variable
Even in finite dimensions this definition is more convenient since it is independent of coordinates. If you are interested in geometric applications this is what you need.
This definition has the ...
- 34.7k
9
votes
Accepted
How is the Gronwall lemma used in this paper?
$\newcommand\al\alpha\newcommand\be\beta\newcommand\la\lambda$The reasoning in the paper is probably as follows:
For real $t\ge0$, letting
$$u(t):=2\la(E|X_t|^2-|EX_0|^2)-1,$$
$$\al(t):=-1+2\la(E|X_0|^...
- 128k
8
votes
Accepted
Show that this process is not a martingale
Here's an approach that comes from
Li, Xue-Mei, Strict local martingales: examples, Stat. Probab. Lett. 129, 65-68 (2017). ZBL1386.60159, https://arxiv.org/abs/1609.00935. Indeed, she mentions this ...
- 29.7k
8
votes
Accepted
Scalar product of random unit vectors
As noted in the comments, by the spherical symmetry, the distribution of the dot product in both parts of your question is the same that of $X\cdot(1,0,\dots,0)$. Moreover, the distribution of $X$ is ...
- 128k
8
votes
Question regarding the Wick tensor in white noise analysis
There is a lot of confusion around the concept of "Wick" product. Much of it is due to the following. As you mention, there is a general formula for the Wick product of a collection of ...
- 10.3k
8
votes
Accepted
Is a martingale conditioned to be large a submartingale?
No.
For a simple counterexample, let's work in discrete time. Consider the following gambling strategy: start with \$0 and bet \$1 on a fair coin flip. If you win, you take your dollar and go home. ...
- 29.7k
8
votes
Different proof techniques of the Atiyah-Singer index theorem
If you want something in lecture notes/book form, the book Stochastic Analysis on Manifolds by Elton Hsu contains a full exposition of the stochastic approach to the Atiyah-Singer index theorem, along ...
- 6,155
7
votes
Does a theory of stochastic differential algebras exist?
Yes. A systematic study of stochastic (differential) algebra could be found in
Grenander, Ulf. Probabilities on algebraic structures. Dover Books, 1981.
Grenander studied the operation of ...
- 8,071
7
votes
Accepted
Geometric characterization of martingales
I do not think Hsu's book is a good place to start with, although it has some strong holds-like details in calculation, neither its depth nor its clarity is comparable to
Stroock, Daniel W. An ...
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7
votes
Why do stochastic integrals depend on the choice of partitioning points?
To complement the excellent answer by Ofer Zeitouni, let me offer a functional analysis perspective. We want to define an integral of the following form: $\int F(W_t)dW_t=\int F(W_t)W'_tdt$, say, for ...
- 8,992
7
votes
What phenomena are better modelled by SDE instead of ODE?
Many biological phenomena can be modelled using SDEs or other stochastic models. For example, disease transmission models which keep track of the numbers of infected $I$ and susceptible $S$ ...
- 218
6
votes
Accepted
How to prove Feller property without using heat kernel estimates
In the case of a diffusion, (1) is implied for example by having bounded coefficients. This follows immediately from applying BDG to $X_t-x$ and doesn't require (2) which is much harder to get.
Note ...
- 10.3k
6
votes
Fundamental Contradiction with Brownian motion
To conclude that something here is wrong, you should be able to indicate a contradiction. Of course, you won't be able to do this, since the two random variables, $M_t:=\sup_{s \in [0,t]} B_s$ and $...
- 128k
6
votes
Walker whose Velocity is a Brownian Bridge
As Kwaśnicki remarked, the velocity process $v_t$ is a Brownian bridge, which can be represented as: $$
v_t = v_0 (1 - \frac{t}{T}) + v_T \frac{t}{T} + (T - t) \int_0^t \frac{1}{T-s} d B_s \;.
$$ (...
- 6,045
6
votes
Accepted
fractional Brownian Motion driven stochastic integrals
Yes, RPT allows you to define a notion of stochastic integral against fBm for a class of integrands that is larger than what Young's theory allows. Assuming that you're really interested in solving ...
- 10.3k
6
votes
Accepted
English translation of "Les aspects probabilistes du contrôle stochastique"
There is no English translation of El Karoui's lecture notes, however her work on Snell envelopes is described in Reflected Solutions of Backward SDE'S, and Related Obstacle Problems for PDE's. For a ...
- 188k
6
votes
Accepted
Filtration exercise
Continuity at zero is included in the RCLL property.
If $X(\cdot,\omega)$ is discontinuous at some $t \in (0,t_0)$, then the left and right limits at $t$ (which exist) must differ. Thus there must ...
- 14.2k
6
votes
Accepted
When are the transition densities of an SDE symmetric?
This question is addressed here. In particular, a diffusion whose transition densities are symmetric is a special case of a $\nu$-symmetric diffusion, and by itself, this symmetry does not uniquely ...
- 6,045
6
votes
A curious martingale
I think Iosif's Fatou lemma argument can be fixed, as follows.
Assume without loss of generality that $X_0 = 0$.
Suppose to the contrary that $X_t \to +\infty$ a.s. Then it must be that $\inf_{t \ge ...
- 29.7k
6
votes
Interpretation of second order term in Fokker-Planck equation
To help the interpretation you may want to rewrite$^\ast$ the Fokker-Planck equation as
\begin{align*}
\frac{d}{dt}p(x)
=& -\nabla\cdot [\tilde{f}p -D\cdot\nabla p],\\
\end{align*}
with $D=\tfrac{...
- 188k
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