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
16
votes
Accepted
Existence of normal number except random numbers
Computable, absolutely normal numbers do actually exist. See
V. Becher, S. Figueira: An example of a computable absolutely normal number, Theoretical Computer Science 270 (2002), 947-958.
- 66.3k
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
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
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
SDE driven by fractional Brownian motion
When $H > 1/2$, one can interpret the integral as a Riemann-Stieltjes integral and one gets solutions by classical Picard iteration in $C^\alpha$ for $1/2 < \alpha < H$. It's not hard to see ...
- 10.3k
8
votes
Accepted
For a stable matrix $B$ and anti-symmetric $T$, such that $B(I+T)$ is symmetric, show that $\mbox{tr}(TB)\leq0$
First let us check that $T$ exists and is unique. Let $\mathrm{Sym}_n$ be the space of symmetric matrices (with real coefficients), $\mathrm{M}_n$ the space of all matrices and $\mathrm{Alt}_n$ the ...
- 96
8
votes
A singular stochastic differential equation
This is sticky reflecting Brownian motion, see for example this relatively recent paper. You can alternatively construct it by taking a reflected Brownian motion and then "stretching out" ...
- 10.3k
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
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
Accepted
How to make sense of recursively defined SPDE solutions, like in Hairer's "Solving the KPZ equation" paper?
What I mean is that
$$
X_\epsilon^\tau(t) = \int_{-\infty}^t P_{t-s} \Pi_0^\perp (\partial_x X_\epsilon^{\tau_1}(s)\, \partial_x X_\epsilon^{\tau_2}(s))\,ds\;,
$$
where $P_t$ denotes convolution with ...
- 10.3k
7
votes
Accepted
Path integral presentation of solutions of Dirac equation
There are several relevant papers:
Path Integral Approach to Relativistic Quantum Mechanics: Two-Dimensional Dirac Equation (1987)
Path Integral for
Relativistic Equations of Motion (1997)
Path ...
- 188k
7
votes
Accepted
A comparison of diffusions
The inequality is not true in general — additional assumptions are needed. I think some kind of monotonicity of $a_1$ and $a_2$ should help, but this is merely a guess.
Here is a counterexample. ...
- 17.2k
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
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
Under what condition we get back path from signatures in rough path theory?
Loops don't get canceled out in the signature. (You might like to compute the signature of a circle. The second iterated integral is nonzero; in fact, by Green's theorem, it gives you the area inside ...
- 29.7k
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
Intuition behind Gubinelli derivative
In a way it is very much like a usual derivative. Recall first that for a regular function $Y$, its derivative $Y'_s$ at a point $s$ is the (unique) number such that
$$
Y_{t,s}=Y'_s(t-s)+ R_{s,t},
$$
...
- 493
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
Accepted
Weighted Lebesgue space with exponential weights: smoothing effect and properties
This kind of weight reminds me a lot of Gaussian weights. Then, you can normalize it and put it into the measure to get an $L^p$ space on $\mathbb R^n$ endowed with the Gaussian measure.
My intuition ...
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 $D=\tfrac{...
- 188k
5
votes
Accepted
Reflecting Brownian motion and its transition probability density
Since building reflected Brownian motion in smooth bounded domains is not a problem, the only potential obstruction to the existence of the transition probabilities is that it escapes to infinity in ...
- 10.3k
5
votes
Why the term "geometric" rough path?
I agree with Martin's answer - but there were other additional and compelling reasons.
A rough path (as in the original papers) was a path in the tensor algebra over V with appropriate "p-...
- 71
5
votes
Parameter estimation for stochastic differential equation from discrete observations
Part 1: Methods
There is no best method, but most methods end up in quite a similar area. Here's an old review which is still quite good, though some of the language and specific methods need to be ...
- 454
5
votes
Accepted
Reference for Feynman-Kac
Please see p.282 of the following ref, it does not call it Feynman-Kac Formula but the whole section 5 is discussing it. It formalized the diffusion using a tensor field over a manifold which is the ...
- 8,071
5
votes
Accepted
Variance of Multi-Dimensional Ornstein Uhlenbeck process
the large-$t$ limit $\rho$ of the variance of $X_t$ is given by
$$\rho=\int_0^\infty \exp(-tH)SS^{\dagger}\exp(-tH^{\dagger})dt$$
where the superscript $\dagger$ indicates the transpose (or the ...
- 188k
5
votes
Tanaka-Meyer formula
It looks to me like nothing's wrong; the local time $\Lambda^Z_t(0)$ of $Z$ at zero is identically zero. This makes a certain amount of intuitive sense, because $Z$ should have "bounded variation at ...
- 29.7k
5
votes
Good papers on stochastic differential equations with applications in finance
For basic theory: Stephen Shreve's books (Stochastic Calculus for Finance I and II) and Martingale Methods in Financial Modelling by Marek Musiela and Marek Rutkowski. Also have a look at Oksendal's ...
- 222
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