All Questions
141 questions
10
votes
2
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Does the strong law of Large Number hold for an infinite dimensional Brownian motion?
For finite-dimensional Brownian motion $W_t$, it is well known that
\begin{equation}
\lim_{t\to \infty}\frac{W_t}{t}=0,\text{ a.s. }\ \ \ \ \hspace{1cm} \langle 1\rangle
\end{equation}
Now suppose we ...
- 121
10
votes
2
answers
2k
views
Covariance function of Brownian motion and the second derivative operator
I recently noticed something about the covariance function of a Brownian motion that I don't quite understand, and I was wondering if anyone could help me.
Suppose $W$ is a Brownian motion, and we ...
- 1,666
9
votes
3
answers
2k
views
When is a continuous path stochastic process be representable as diffusion or Ito process?
When can a continuous path (Markovian) stochastic process in one dimension be represented as an Ito or a diffusion process? What are the examples when it can not be?
- 2,239
8
votes
2
answers
1k
views
The Wiener measure of an open set
There is so much written about the Brownian motion and I suspect the answers to the questions below are hidden in somewhere in the literature but I cannot find them
Denote by $E$ the Banach space ...
- 34.7k
8
votes
2
answers
422
views
Regularity of translations for Brownian motion
Let $B_t$ be the classic Brownian motion. I understand that, if $s>1/2$, almost surely $B_t$ is nowhere $s$-Hölder continuous i.e. almost surely for no point $x$ it happens that $B_t\in C^s(x)$.
...
- 319
8
votes
1
answer
568
views
Escape Time of Fractional Brownian Motion
Let $B(t)$ be Brownian motion with $B(0)=x>0$ and let $A>x$. It is well known that the expected time for $B(t)$ to escape the interval $[0,A]$ is equal to $x(A-x)$.
Is the expected time known ...
- 3,626
7
votes
1
answer
875
views
White noise vs. black noise
In this excellent lecture ("2d Percolation Revisited") Stanislav Smirnov mentioned the connection of the theory of percolation with the notion of the so called black noise—see at 29:42 (the notion ...
- 9,330
7
votes
2
answers
984
views
Brownian motion in $n$ dimensions
Consider a particle starting at the origin in $\mathbb{R}^n$ and undergoing Brownian motion. Is there an expression known for the probability of the particle hitting the sphere $S^{n - 1}_r = \{x \in \...
- 73
7
votes
2
answers
5k
views
Properties of the time integral of Wiener process
Let $W_t$ be a Wiener process and consider the time integral
$$ X_T:= \int_0^T W_t dt $$
It is often mentionend in literature that $X_T$ is a Gaussian
with mean 0 and variance $T^3/6$.
I am ...
- 2,810
7
votes
1
answer
4k
views
Change of time variable in Wiener process
I'm following a solution of an SDE from here
http://www.math.ethz.ch/~delbaen/ftp/preprints/CEV.pdf
Start with the SDE
$$
dX_t = \delta dt + 2\sqrt{X_t} dW_t
$$
consider a deterministic time change
$...
- 173
7
votes
2
answers
613
views
Fractional Brownian motion of Riemann-Liouville type is not a semimartingale
Given a filtered probability space $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$ satisfying the usual conditions, $B$ a standard one-dimensional Brownian motion and $H\in(0,1/2)$. Consider the process $...
- 199
7
votes
1
answer
1k
views
Moment bounds on exponential martingale
Consider the exponential martingale used in the Girsanov transformation of
measure:
$$Z(t) = \exp\Big(\int_0^tXdW - \frac{1}{2}\int_0^t|X|^2ds\Big)$$
so that $Z$ solves the sde $dZ = ZXdW$ where $W$ ...
- 143
7
votes
1
answer
467
views
Properties of the algebraic self-difference set of Brownian motion zeros
As I was trying to exhibit new interesting(?) path transformations of Brownian motion, I became interested in
the (random) set of times $t$ such that $B(t)=B(t+1)=0$, where $B(t)$ denotes a standard ...
- 704
7
votes
2
answers
307
views
PDE for the probability of Brownian motion staying in an area (reference request)
I am looking for a (preferably some monograph) reference on the following fact:
$$
u ( t, x ) = \mathbb{P} \{ x + B_s \in A \ \text{for all} \ s \leq t \}
$$
satisfies the heat equation
$$
\frac{\...
- 620
7
votes
1
answer
278
views
A Converse of the Skorokhod Embedding Theorem
I am wondering whether the following "sort of converse" of Skorokhod's embedding theorem holds:
Suppose that $\{D_t\}_{t \geq 0}$ is a stochastic process with continuous paths, $D_0 = 0$, and suppose ...
- 241
6
votes
1
answer
579
views
Is this a Brownian motion?
I am building a 2D stochastic process as follows. I start with a point $P_0=(0,0)$. Then $P_k=(X_k,Y_k)$ is defined as follows, for $k>0$:
\begin{align}
X_k & =X_{k-1}+R_k \cos(2\pi\theta_k) \\
...
- 3,098
6
votes
1
answer
2k
views
Brownian motion and its maximum and its minimum
Let $W_u, 0\leq u \leq t$ be Brownian motion.
Let $m_t= min_{0\leq u\leq t} W_u$ and $M_t = max_{0 \leq u \leq t} W_u$.
The fact that $(M_t , W_t)$ is absolutely continuous with respect to Lebesgue ...
- 161
6
votes
1
answer
374
views
Large deviation for Brownian path on $[0,\infty)$
It seems strange to me that all we can find about Schilder's theorem in the literature is on a finite interval of Brownian path.
If we equip the space of continuous function starting from $0$, ...
- 123
6
votes
1
answer
608
views
weak convergence of the solutions to stochastic heat equation
$W(t,x)=\sum_ic_ie_i(x)B^i_t$ is a Brownian motion in $L^2(R^d)$, where $\{e_i\}$ is the standard orthogonal basis and $\sum_ic_i^2<\infty$.
$$\partial_t u(t,x)=\Delta u(t,x)+u(t,x)\dot{W}(t,x)$$
...
- 61
6
votes
1
answer
133
views
Coupling/Ordering of Brownian bridges
Suppose I have two 1D Brownian bridges $(B^{(1)}_t,t\in [0,1]),(B^{(2)}_t,t\in [0,1])$, one from $0$ to $0$ and one from $x$ to $y$ where $x,y \geq 0$. Is there a neat way to show that there exists a ...
- 228
6
votes
0
answers
292
views
Running maximum/supremum of Brownian motion: add information to make it a Markov process?
Let $B_t$ be standard Brownian motion, and let $M_t = \sup_{0 \leq s \leq t} B_s$ be its running maximum. $M_t$ is not a Markov process, but we can augment it with additional information to make it ...
- 398
6
votes
0
answers
220
views
Reference request: Stochastic integration and martingale theory on the whole real line
I'm looking for a thorough treatment of stochastic integration and/or martingale theory on the whole real line, i.e. a way to construct a Brownian motion $(B_s)_{s \in \mathbb{R}}$ (if a two-sided BM ...
- 617
5
votes
3
answers
1k
views
"Practical" use of time-continuous stochastic processes like Wiener process or Poisson (point) process?
If one uses the Wiener process as an ingredient to model something, then for practical purposes one could just as well take a simple discrete random walk (with sufficiently fine scale).
If one uses a ...
- 59
5
votes
2
answers
688
views
Endpoint of Brownian motion conditional on high maxima
Note: This question is closely related to an earlier question: A large noise limit.
Let $W$ be a standard one dimensional Brownian motion.
For every $\varepsilon > 0$, let $A_\varepsilon$ denote ...
- 6,215
5
votes
2
answers
185
views
Density near at $0$ for the integral of the positive part of the Brownian motion
This question was asked recently on MO and then deleted by the owner, user Aalon. I think the question deserves to be answered, which is what I will try to do here. Aalon was reading this paper, where ...
- 128k
5
votes
1
answer
548
views
Largeness of the set of zeroes of a Brownian motion
Definitions:
A measurable subset $S$ of $\mathbb R$ is said to be mesoscopic if there exists a continuous function $f: \mathbb R \to \mathbb R$ such that $f(S)$ is Lebesgue measurable and has nonzero ...
- 6,215
5
votes
2
answers
725
views
Brownian motion in $\mathbb{R}^n$, probability of hitting a set
Consider a particle undergoing Brownian motion in $\mathbb{R}^n$, starting at the origin, and let $B(t)$ denote its position at time $t$. Let $X$ be an arbitrary subset of $\mathbb{R}^n$. I am trying ...
- 53
5
votes
2
answers
289
views
Bounding Brownian motion and an Ito process simultaneously
Let $(W_t)_{t\geq0}$ be a standard Brownian motion in $\mathbb{R}^n$ and $(A_t)_{t\geq0}$ be an adapted matrix-valued process such that $A_t$ is a positive symmetric matrix with bounded operator norm :...
- 245
5
votes
1
answer
2k
views
Blumenthal and Kolmogorov 0-1 law
Blumenthal's 0-1 law see theorem 5.8/5.9 tells us that an event in the germ $\sigma-$ algebra has either probability zero or one with respect to a measure induced by a Brownian motion starting in some ...
- 111
5
votes
1
answer
284
views
Malliavin derivative of stopped Brownian motion
Cross-posted from: "https://math.stackexchange.com/questions/3917971/malliavin-derivative-of-stopped-brownian-motion"
I have a small question concerning the Malliavin derivatives. It could ...
- 393
5
votes
1
answer
523
views
Scaling of First-passage times for Random Walk on integer lattices
Consider simple symmetric random walk $S_{n} = (S_{n}^{(1)},\dots,
S_{n}^{(d)})$ on the d-dimensional integer lattice with starting point the origin.
Let $\tau_{N}$ be the first time $S_{n}$ exits ...
- 61
5
votes
0
answers
653
views
Explicit martingale representation for a Brownian bridge
Let $W$ denote a Wiener process, $\displaystyle M_t = \max_{0 \le s \le t} W_s$ its running maximum. The martingale representation of $M$ is known explicitly:
$$M_T = \sqrt{\frac{2T} \pi} + \int_0^T ...
- 341
4
votes
2
answers
688
views
Supremum of difference of Brownian bridges: strictly positive wp 1?
EDIT: the original $\ge$ is now $>$ (sorry for the typo!)
Let $B_1(\cdot)$ and $B_2(\cdot)$ denote independent, standard Brownian bridges, i.e., they are mean-zero Gaussian processes on $[0,1]$ ...
- 143
4
votes
1
answer
447
views
Area enclosed by Brownian motion (without winding number)
The question Average Value of Area Closed by Brownian Motion turned out to be about the Lévy area process, which measures "signed area with multiplicity" enclosed by Brownian motion (e.g. each ...
- 29.8k
4
votes
1
answer
773
views
SDE-removal of the diffusion coefficients
from math.stackexchange
I'm currently looking at stochastic differential equations with irregular coefficients such as $W^{1,p}_{loc}$. If I have
\begin{align}
dX_t=b(X_t)dt+\sigma dW_t,
\end{align}
...
- 41
4
votes
1
answer
509
views
Conditional stochastic integration
Let's say we have two functions $h(s)$ and $g(s)$. We can easily simulate a stochastic integral, e.g.
$$t \mapsto \int_0^t h(s) dB(s) \sim \mathcal{N}\bigg(0, \int_0^t h(s)^2 ds \bigg). $$
What is the ...
4
votes
1
answer
143
views
Reflecting Brownian motion in disk
What is the transition density function of a reflecting Brownian motion in $\mathbb D \overset{\mathrm{def}}= \{z \in \mathbb C : \lvert z\rvert < 1\}$ and how to compute it?
The transition density ...
- 177
4
votes
2
answers
456
views
Converse of Itô's formula
Let $f,h,g$ be continuous functions and $B$ a real Brownian motion. We suppose that a.s. $$\forall u \in \mathbb{R}_+,f(B_u)=f(B_0)+\int_0^ug(B_r)dB_r+\frac{1}{2}\int_0^uh(B_r)dr.$$
Prove that $f$ is ...
- 573
4
votes
1
answer
404
views
Weighted global Holder property for Brownian motion paths
It is well-known that the Brownian motion (Wiener process) is almost sure locally $\alpha$-Holder for any $\alpha<1/2$. That is, with probability 1
$$
\sup_{t,s\in[0,1]}\frac{|W_t-W_s|}{|t-s|^{\...
- 931
4
votes
1
answer
2k
views
Expectation of the time t standard brownian motion stopped at itself's square
I have a one dimensional standard brownian motion $W$ defined under a stochastic basis with probability $\mathbf{Q}$ and filtration $\left(\mathscr{F}\right)_{t\in{\mathbf{R}}_{+}}$, and I want to ...
- 255
4
votes
0
answers
127
views
A "resampling identity" for the Bessel(3) process
I've come across the following resampling identity and was wondering if this is known since it seems rather natural. Take $X$ a two-sided Brownian motion conditioned to always stay below $1$. (So if ...
- 10.3k
4
votes
0
answers
167
views
Occupation time of SDE
Let $b:\mathbb{R}^d\to\mathbb{R}^d$ be locally Lipschitz and assume that, for any $x\in\mathbb{R}^d$ and any $f\in C^{\infty}([0,1],\mathbb{R}^d)$, the equation
$$
X_t^{x,f}=x+\int_0^t b(X_s^{x,f})\,...
- 93
4
votes
0
answers
129
views
Tail for the integral of a diffusion process
I would like to compute the following tail,
$$
\mathbb{P}\left(\int_{0}^{T} f(X_t)\mathrm{dt}>x\right),
$$
assuming
$$
\mathbb{P}[f(X_t)>x] = x^{-\alpha} \log(x),
$$
and $X$ is a diffusion ...
- 41
3
votes
2
answers
490
views
SDE driven by fractional Brownian motion
Let $B^H$ be a fraction Brownian motion of Hurst parameter $H$. Consider the SDE driven by $B^H$ as below:
$$dX_t = b(t,X_t)dt + a(t,X_t)dB^H_t,\quad \forall t\ge 0.$$
I am looking for references that ...
- 1,334
3
votes
1
answer
933
views
Brownian motion - probability of striking a sphere in $\mathbb{R}^n$ (a clarification)
This is primarily in reference to this question on MO. Serguei Popov's answer gives an explicit formula for the probability of a Brownian particle starting at the origin in $\mathbb{R}^n$ hitting the ...
- 33
3
votes
2
answers
3k
views
Quadratic variation for discrete Martingale
Is there any analogue of continuous martingale quadratic variation for the discrete case? If so, are there any theorems which characterize simple random walk using quadratic variation - similar to ...
- 133
3
votes
1
answer
467
views
Generator of Wiener process and its running maximum
This was originally posted on Math StackExchange a long time ago, but got no answer (even after a bounty).
Let $W$ be a standard linear Wiener process issued from zero and $M$ its running maximum
$$
...
- 279
3
votes
1
answer
903
views
Exercise on a hitting time for a Brownian Motion
I'm following Chapter 3 of "Brownian Motion", by Peres and Mörters, about The Dirichlet Problem(DP). As it is known, in order to obtain existence and uniqueness of a solution for DP it is necessary to ...
- 203
3
votes
1
answer
655
views
Forgery theorem: the Brownian motion stays close to any curve with positive probability
In a paper I am reading the authors claim that, if $B$ is a standard BM in $\mathbb{R}$ and $f\in C([0,1],\mathbb{R})$, then for any $\epsilon>0$
$$
\mathbb{P}(\sup_{t\in [0,1]}|B_t-f(t)|<\...
- 1,149
3
votes
1
answer
229
views
How to prove excursion process is a Poisson point process?
This question comes from book Ju-Yi Yen and Marc Yor P59 and P60,
On page 59, "Define $\mathcal{Z}_\omega=\{t:B_t(\omega)=0\},$ and $\tau_l$ is the inverse local time. The complement of $\mathcal{...
