17 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 ...
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 $...
14 votes

Intuition and/or visualisation of Itô integral/Itô's lemma

I found this explanation somewhere and wrote it down in my personal notes. I will explain with an example that I think exemplify why Riemann-Stieltjes will provide the wrong answer. First, let's ...
  • 325
9 votes

Analytic Solution to SDEs

Yes. Unexpected weak solutions to the SDE $$ d Y = - \Phi'(Y) dt + \sqrt{2} dW \quad Y(0) \in \mathbb{R} $$ are available. To see this, transform the associated Fokker-Planck equation into a Schr&...
9 votes
Accepted

Brownian motion in $n$ dimensions

The process $\|B(t)\|$ is called $n$-dimensional Bessel process (or Bessel process with parameter $\nu=\frac{n}{2}-1$). I think formula $\bf 4$.1.1.4 of Borodin-Salminen "Handbook of Brownian Motion -...
9 votes

Intuition about Skorohod integral

Unfortunately, calling the Skorohod integral an "integral" is a bit of a misnomer, as it doesn't really have many of the properties which you would naturally associate with integrals, except for the ...
9 votes
Accepted

Brownian motion - probability of striking a sphere in $\mathbb{R}^n$ (a clarification)

Your intutive reasoning is leading you astray because you are thinking of Brownian motion as behaving like a smooth curve, for which there is a well-defined "direction" in which it is heading. ...
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 $\alpha B_\cdot$, ...
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 ...
8 votes
Accepted

Change of time variable in Wiener process

The time change described in the question may be handled as follows. Recall that if $W(t)$ is a standard Brownian motion then $$ W(\tau(b))-W(\tau(a)) $$ has the same distribution as $$ \int_a^b \...
8 votes
Accepted

Average Value of Area Closed by Brownian Motion

One way to define the "enclosed area" for a curve $\mathbf{r}(t)$ in the $x$-$y$ plane of duration $T$, with $\mathbf{r}(0)=\mathbf{r}(T)$, is via the socalled algebraic area $A=\tfrac{1}{2}\int_0^T (\...
8 votes
Accepted

Can all local martingales be represented using only Brownian motion and finite variation processes?

First, a martingale is always only specified with respect to a filtration, and so is thus a local martingale. You do not specify any filtration in your problem, so I assume you mean the natural ...
8 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 ...
  • 1,463
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 ...
8 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 ...
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 ...
  • 7,593
7 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 ...
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 ...
  • 6,344
7 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 ...
6 votes

Intuition and/or visualisation of Itô integral/Itô's lemma

One way to improve intuition is to work out a couple of Discrete versions of Ito's lemma Øksendal (6th edition) Example 3.1.9: almost surely, $$ B_t^2 - t = \int_0^t 2B_s dB_s $$ This has a ...
6 votes

A version of Wald identity

OK, since I slipped again (there is no way I will believe nowadays that after 40 one's abilities do not decline sharply, though when I was younger I was more optimistic about the lifetime of a ...
  • 54.4k
6 votes

Average Value of Area Closed by Brownian Motion

There is a beautiful connection between the area swept out by Brownian motion and the Dirichlet function: $L(s)=\sum_{n=0}^{+\infty} \frac{(-1)^n}{(2n+1)^s}$. Proposition: Let $A_t$ be the area ...
6 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 ...
  • 7,593
6 votes
Accepted

How to calculate the PSD of a stochastic process

You ask for the spectral analysis of a nonstationary stochastic process. Because the autocorrelation function $C(s,t)$ now depends on the two times $s$ and $t$ separately, and not only on their ...
6 votes

Asymptotic behavior of an integral of OU process

Using the notation of the OP, let $I(t)=\int_0^t X_s^2 ds$ where $X$ solves the above SDE. By Chebyshev's inequality, we have that $$ P(I(t) > \alpha \mid X_0 = x) \le \frac{E\left\{ I(t) \mid X_0 ...
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 $...
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 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 ...
5 votes
Accepted

Stochastic methods for solving very high-dimensional PDE

It seems to me this question "has not received enough attention" because of the conflation of two issues: dimensional reduction of a high-dimensional PDE and stochastic (Monte Carlo) integration of ...
5 votes
Accepted

Origins and Industrial Applications of stochastic processes (eg. Brownian motion) on Riemannian manifolds

The earliest "industrial" application I know is in the context of microwave engineering: the eigenvalues of the transmission matrix through a waveguide with random scatterers perform a Brownian motion ...

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