17 votes
Accepted

Why the term "geometric" rough path?

Geometric rough paths have the property that if you want to solve an equation with values in a manifold, choose a coordinate chart, and write in local coordinates $$ dY^i = V_0^i(Y)\,dt + \sum_j V_j^i(...
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 ...
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.
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 $...
11 votes
Accepted

Does Brownian motion immediately visit both sides of a Jordan curve?

As to question 2: Planar Brownian motion started at $y_0$ will almost surely loop around $y_0$, i.e., disconnect $y_0$ from $\infty$ immediately, so it has to hit $A$ and $B$ immediately, too, and $\...
10 votes
Accepted

What is the idea behind interpolation spaces?

In the definition, "diagonal" does not make much sense, you probably mean "self-adjoint", although this can be relaxed to "m-sectorial" (powers of $A$ still make sense then). Also, it is irrelevant ...
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

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 ...
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

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
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
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 ...
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" ...
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

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 ...
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 ...
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. ...
6 votes

Well-posedness of Fokker-Planck equation

The argument in JKO (short for Jordan-Kinderlehrer-Otto-98) is based on a Lyapunov function for the Fokker-Planck equation. As such, it requires that this Lyapunov function evaluated at $\rho^0$ be ...
6 votes
Accepted

When does the cumulative distribution function solve the Kolmogorov backward equation?

If the function $f : \mathbb{R} \to \mathbb{R}$ is discontinuous, then $u(t,x)= \mathbb{E}_xf(X(t))$ may not satisfy the initial condition, in the sense that the limit statement: $$ \lim_{(t,s) \to (0^...
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

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

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

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 ...
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

Solve SDE $dX_t=(c+\sigma_\zeta W'_t)X_tdt + \sigma_\epsilon dW_t$

Using a (random) integrating factor, the solution is explicitly: $$ X_t =\exp\left( \sigma_{\zeta} \int_0^t W'_s ds \right) \times \left[ x_0+\int_0^t \exp\left(- \sigma_{\zeta} \int_0^s W'_r dr \...
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

Does there exist a stochastic time derivative?

I think the chain rule $d[f(Z_t)]=f'(Z_t)\circ dZ_t$ is valid when the product $\circ$ is defined as in Stratonovich stochastic integral (while the SDE uses Ito's). Note that $g(Z_t)\circ dt$ doesn't ...
  • 3,035
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 ...
5 votes
Accepted

Good papers on stochastic differential equations with applications in finance

As indicated in the comments, the field is very wide, but I understand from the comment of the OP to zab's answer that there is a specific interest in the more narrow subtopic of applications of ...
5 votes

Is this a "contradiction" on stochastic Burgers' equation? How to understand it?

It is not true that the bound $dM/dt = -\beta M^2$ implies that $M$ blows up almost surely. For example, with $b = 2$, there is a non-zero probability that $\beta < ce^{-t}$ for all $t>0$, for ...

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