All Questions
Tagged with pr.probability brownian-motion
247 questions
-2
votes
0
answers
52
views
Density of squared bessel process
I was trying to find a transition density function for a squared Bessel process. In the book "Continuous martingale and Brownian motion" by Revuz and Yor, I find a Corollary on page 441 that ...
- 11
4
votes
1
answer
66
views
Expectation bounds on supremum of family of martingales
Suppose I fix a filtered probability space $(\Omega, \mathcal{F}, \mathbb{F}, P)$ and on it a Brownian motion $B$. Let $\tau_\alpha$ denote a set of stopping times which satisfies $\sup_\alpha \tau_\...
- 143
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
5
votes
0
answers
68
views
Distribution of this integral of Fourier multiplier
In Barashkov and Gubinelli (2019) section 2, the authors make the claim that the distribution of $$Y_t = \int_0^t \langle D \rangle^{-1}\sigma_s(D)dX_s$$ is given by the pushforward $(\rho_t(D))_*\...
- 51
2
votes
0
answers
158
views
Conformally mapping between the upper half complex plane, and the plane with a tree on spatial points removed
A stochastic process such as SLE$_{\kappa}$ can be defined by taking the scaling limit of a curve in the upper half complex plane: put simply, one removes a line segment, then another, $n$ times, each ...
- 640
2
votes
0
answers
61
views
Characterisation of Bessel process
Let $\delta \in (0, 2)$; $(X_t)_{t \ge 0}$ a nonnegative continuous Markov process. Suppose that
For each $T \ge 0$, if we write $\tau \overset{\mathrm{def}}= \inf\{t \ge T : X_t = 0\}$, then $(X_{T +...
- 177
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
2
votes
0
answers
154
views
Kolmogorov complexity of Brownian motion
Given a 2 dimensional Brownian motion path $B\subset \mathbb{R}^2$, I am trying to show that for any oracle $A\subset\mathbb{N}$, there is a point $x \in B$ such that
$$\liminf_{r\to\infty}\frac{K_r^A(...
- 21
3
votes
0
answers
60
views
Comparison theorem for SDEs driven by a continuous martingale
Consider the well-known comparison theorem for SDEs, versions of which appear in several textbooks, e.g., Karatzas and Shreve, Proposition 5.2.18, or Revuz and Yor, Theorem IX.3.7.
The result states ...
- 103
7
votes
2
answers
307
views
PDE for the probability of Brownian motion staying in an area (reference request)
I am looking for a (preferably some monograph) reference on the following fact:
$$
u ( t, x ) = \mathbb{P} \{ x + B_s \in A \ \text{for all} \ s \leq t \}
$$
satisfies the heat equation
$$
\frac{\...
- 620
1
vote
0
answers
133
views
A question about one Malliavin derivative calculation
Recently, I've asked here a question. While trying to find an answer on my own, I found an idea which I now will briefly describe below. I am not familiar enough with the Malliavin calculus, so my ...
- 620
1
vote
0
answers
99
views
Expectation of $B_u \operatorname{argmax}_t B_t$
This question is a repost from math.stackexchange. The question turned out to be harder than I initially thought, so I decided to try my luck here.
Yesterday I asked a question about the joint law of ...
- 620
7
votes
1
answer
524
views
What happens when the diffusion term in an SDE becomes zero?
Consider this time-homogeneous SDE, in the Ito sense:
$$dX_t= -(X_t-a)\,dt+\sigma(X_t)\,dW_t,$$
where $W_t$ is standard Brownian motion, $a<b\in\mathbb{R}$, $X_0\leq b$ a.s., and $\sigma(b)=0$. ...
- 103
1
vote
1
answer
101
views
Reference for the 'Brownian Representation Formula'
I am reading a paper ('Hydrodynamics of the N-BBM Process', by De Masi, Ferrari, Presutti, Soprano-Loto) which quotes the 'Brownian representation formula' to represent the solution of a free boundary ...
- 177
4
votes
0
answers
142
views
Algebraic area of Brownian half-plane excursion
Is anything known about the distribution of the algebraic area, à la Lévy's stochastic area, of a Brownian excursion in the half-plane? To be precise, letting $x>0$, we consider the path $(X_t,Y_t)...
- 3,927
3
votes
1
answer
314
views
Laplace transform of Brownian motion functional
Let $(B_r,r\geq 0)$ be a standard Brownian motion on $\mathbb{R}$ started at $0$. I am interested in the quantity
$$g(s,t) = \mathbb{E}_0\left[ \exp \left(- \beta \int_s^t \left\vert \frac{B_r}{r}\...
- 228
3
votes
1
answer
180
views
Are the paths of the Brownian motion contained in a suitable RKHS?
Let $H_B$ be the reproducing kernel Hilbert space (RKHS) of the Brownian Motion $(B_t)$ on $[0,1]$. It is well known that with probability 1 the paths of $(B_t)$ are not contained in $H_B$.
But is ...
- 31
3
votes
0
answers
143
views
Stochastic braids
I am definitely not a probability guy, but I'd like to have a heuristic answer to the following question: do $n$ independently moving points in an open, connected, bounded region $R$ tend to "...
- 2,152
4
votes
1
answer
143
views
Reflecting Brownian motion in disk
What is the transition density function of a reflecting Brownian motion in $\mathbb D \overset{\mathrm{def}}= \{z \in \mathbb C : \lvert z\rvert < 1\}$ and how to compute it?
The transition density ...
- 177
2
votes
0
answers
47
views
From large deviations to finite time probability tails
Cross-Post from Math.SE
Let $(B_t)$ be a standard $d$-dimensional Brownian motion. It is well-known that
$$\mathbb P(\sup_{s\in[0,t]}|B_s|\ge \alpha) \le 4de^{-\alpha^2/2dt}.$$
One possibility to ...
- 245
3
votes
2
answers
490
views
SDE driven by fractional Brownian motion
Let $B^H$ be a fraction Brownian motion of Hurst parameter $H$. Consider the SDE driven by $B^H$ as below:
$$dX_t = b(t,X_t)dt + a(t,X_t)dB^H_t,\quad \forall t\ge 0.$$
I am looking for references that ...
- 1,334
1
vote
0
answers
134
views
Generating realizations from $n$-dimensional geometric Brownian motion where the variables are constrained to sum to 1
Is there a way to simulate an $N$-dimensional geometric Brownian motion i.e. variable $$x_i, i \in [1, N] $$ is diffusing in log-space such that $$\log (x_i)$$ follows a Brownian motion with a given ...
- 21
2
votes
0
answers
164
views
Fractional Brownian motion covariance with a twist
Let $H \in (0, 1)$, $D \in \mathbb{R}$ and assume that the following function
$$
r ( t, s ) = \frac{1}{2} \, \Big[ t^{2H} + s^{2H} - | t - s |^{2H} \Big] + D \, t^H s^H,
\quad t, \, s \geq 0
$$
is ...
- 620
0
votes
0
answers
95
views
Prove that $\forall x,y \in \mathbb{R}^d , P_x\{y\in B\mathopen]0,1]\}=0$
I'm folowing the proof of corollary 1.8 page 5 of Mörters - Sample path properties of Brownian motion.
I want to show that $$\forall x,y \in \mathbb{R}^d , P_x\{y\in B\mathopen]0,1]\}=0$$ where $B$ is ...
- 11
2
votes
0
answers
66
views
Joint tail for Brownian motion $P[B_{t_1}>g_1,...,B_{t_n}>g_n]$
Maybe not surprisingly there seems to be a lack of in-depth study of sharp estimates for the joint tail of Brownian motion over different times
$$P[B_{t_1}>g_1,...,B_{t_n}>g_n]$$
for strictly ...
- 5,474
2
votes
1
answer
273
views
If $u$ is harmonic, $\exists \alpha,\beta \in \mathbb{R},\forall x\in \mathbb{R}^d,u(x) \leq \alpha |x|+\beta,$ then $u$ is affine
We consider a harmonic function $u:\mathbb{R}^d \to \mathbb{R}$ $(\Delta u=0).$ Suppose that $$\exists \alpha,\beta \in \mathbb{R},\forall x\in \mathbb{R}^d,u(x)\leq \alpha |x|+\beta.$$
Therefore $u-u(...
- 573
2
votes
1
answer
178
views
Is every simply connected domain regular?
Recall that a domain $D \subseteq \mathbb C$ is called regular if for each point $x \in \partial D$, we have $\mathbf P_x\lbrack \tau_D = 0\rbrack = 1$, where $\tau_D = \inf\{t > 0 : B_t \notin D\}$...
- 177
1
vote
1
answer
100
views
Characteristic exponent after Girsanov transformation
Let $B$ be a standard Brownian motion. Its characteristic exponent (or Fourier transform) is easily calculated to be
$$ \mathbb E [e^{ixB_t}] = e^{-\frac 12 x^2 t}. $$
Now I want to apply a Girsanov ...
- 245
7
votes
2
answers
613
views
Fractional Brownian motion of Riemann-Liouville type is not a semimartingale
Given a filtered probability space $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$ satisfying the usual conditions, $B$ a standard one-dimensional Brownian motion and $H\in(0,1/2)$. Consider the process $...
- 199
2
votes
0
answers
282
views
Identify two continuous martingales in law as time-changed Brownian motions
Let $W$ be a Brownian motion and $\alpha$ be a progressively measurable process taking values in $\mathbb R_+$. Set $\beta_t:=\max(\alpha_t, 1)$ for all $t\ge 0$. Define respectively $X$, $Y$ by
$$X_t:...
- 1,399
2
votes
1
answer
291
views
Joint distribution for sticky Brownian motion
$\newcommand{\R}{\mathbb R}$The one-dimensional Sticky Brownian Motion (SBM in short) is an $\R$-valued Markov process given by
\begin{gather*}
dX_t=1_{[X_t\neq 0]}dB_t\\
L_t(X)=\int_0^t 1_{[X_s=0]}ds,...
- 5,380
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
1
vote
1
answer
103
views
Brownian motion hitting open set starting from its boundary
Let $\{W(t),\,t \in [0,1]\}$ be a standard Brownian motion in $\mathbb{R}^d$, starting from $0$. Let $U$ be a non-empty open set such that $0 \in \partial U$.
Which conditions on $U$ are necessary and ...
- 177
3
votes
0
answers
180
views
Wiener sausage of a Brownian motion with coordinates scaled differently
The Wiener sausage of a standard Brownian motion $\{W(t),0 \leq t \leq T\}$ in $\mathbb{R}^2$ is the set $S(T,R)=\bigcup_{0 \leq t \leq T} W(t)+B(0,R)$, where $B(x,r)$ denotes a ball in $\mathbb{R}^2$ ...
- 177
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
0
votes
1
answer
163
views
Stability of SDE fBM
Consider an n-dimensional Ito process
$$
X_t^x = x + \int_0^t\, \alpha(s)ds + \int_0^t\,\beta(s)\,dB^H(s),
$$
where $1/3<H<1$ is the Hurst parameter for an $n$-dimensional fractional Brownian ...
2
votes
2
answers
131
views
Density of $W_t$ assuming it stayed above a line $L$
Let $W_t$ be a Wiener process with $W_0=0$, and let $L=\{at+by=c\}$ be a line with $c/b<0$ (i.e. the line crosses the $Y$-axis below $0$).
Assume that $W_t$ stayed above $L$ up to time $T$. What is ...
- 2,788
1
vote
1
answer
245
views
Stochastic integral with non-anticipating integrand
Let $B$ be a Brownian motion. We want to define $$ \int_{0}^{t} B_{s} dB_{s} : = \lim_{n \to \infty } \sum_{k = 1}^{2^{n}t} B_{\frac{k-1}{2^{n}}}[ B_{\frac{k}{2^{n}}} - B_{\frac{k-1}{2^{n}}}]. $$
To ...
- 131
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
3
votes
1
answer
578
views
Quadratic variation of supremum of brownian motion
I would like to know if in some book or how could I compute the quadratic variation of the supremum of the bronian motion $S_t=\sup_{s\in[0,t]}W_s$ where $W$ is a Brownian motion. I was thinking ...
- 133
2
votes
1
answer
294
views
What is the quadratic variation of $W(B(t))$?
Let $W$ be a two sided real valued Brownian motion. Let $B$ be a one sided Brownian motion independent of $W$. Consider the process $X(t)=W(B(t))$. Is the quadratic variation finite and if it is, what ...
- 1,904
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 $\epsilon>0$
$$
\mathbb{P}(\sup_{t\in [0,1]}|B_t-f(t)|<\...
- 1,149
1
vote
1
answer
215
views
Bernoulli trials with small dependencies: asymptotics (central limit theorem, law of the iterated logarithm)
Let $\{X_k\}$ be a sequence of random variables, with $X_k\in\{+1, -1\}$ for $k>0$, generated as follows.
First, define $S_n=X_1+\dots +X_n$, with $X_0=S_0=0$, and let $0<\beta<\frac{1}{2}$.
...
- 3,098
0
votes
1
answer
211
views
Step in proof of Itô formula
I am reading a book on stochastic processes. The author proved Itô formula for $f(t,w(t))$ where $w(t)$ is brownian motion with filtration $F_t$. Then he wants to prove Itô formula for $x(t)=a(t)+b(t)...
0
votes
1
answer
152
views
Does the convergence of $f_n$ imply the convergence of $\mathbb P[\inf_{0\le s\le t}(W_s-f_n(s))\le 0]$?
Let $(f_n)_{n\ge 1}$ be a sequence of non-decreasing and continuous functions defined on $\mathbb R_+$ and taking values in $[0,1]$. Further, for each $t\ge 0$, $n\mapsto f_n(t)$ is non-decreasing. ...
user478657
0
votes
1
answer
129
views
Is this set negligible?
Let $(W_t)_{t\ge 0}$ be a standard Brownian motion starting at zero. Let $f: [0,1]\to\mathbb R$ be a function that is righ-continuous with left limits. Set
$$A:=\left\{\omega\in\Omega: \inf_{0\le t\le ...
user478657
2
votes
1
answer
182
views
Mean of log-normal variable when exponent is replaced by runnung maximum of Ito-integral
Let $W=\{W_t\}_{t\in[0;1]}$ be a real-valued Brownian motion, $\{F_t\}_{t\in [0;1]}$ the filtration generated by $W$, augmented with the nullsets. Let $\{\sigma_t\}_{t\in[0;1]}$ be a continuous and ...
- 335
1
vote
1
answer
159
views
Conditional probability distribution of a Brownian particle surviving forever
Consider the drift Brownian motion $X_t:=1+bt+W_t$, where $(W_t)_{t\ge 0}$ is a Brownian motion starting at zero. Set $\tau:=\inf\{t\ge 0: X_t=0\}$. Assume $b>0$, then $\mathbb P[\tau=\infty]>0$....
- 1,334
1
vote
0
answers
105
views
Book: Continuous martingale and Brownian motion
I am reading the book "continuous martingale and Brownian motion" 1995_Revuz. It reads the following proposition 3.2 in Chapter VII. That confused me a lot. Where $T_r, T_l$ is the hitting ...
1
vote
1
answer
139
views
Characterization of Brownian motion: processes with right-continuous paths
I am looking for a reference with a proof for the following fact:
If a right-continuous martingale $(X_r)_{ r \geq 0}$ is such that $X_0=0,(X^2_r-r)_r,(X_r^3-3rX_r)_r,(X_r^4-6rX_r^2+3r^2)_r$ are ...
- 573