Questions tagged [brownian-motion]

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the distribution of a stopping time of a Brownian motion

Is there example of a stopping time of a standard brownian motion which has discontinuous distribution? is there any general result for such stopping time?
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4 votes
1 answer
136 views

Solution of SDE at finite time, continuity of pdf

I'm looking at the Langevin dynamics described by the following SDE $$d X_t = - \nabla U(X_t) \, d t + \sqrt {2 \Sigma} \, d B_t,$$ where $X_t \in \mathbb R^d$, $\nabla U(\cdot)$ has some regularity ...
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2 votes
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Brownian motion reflected at a trailing barrier

Let $X_t$ be a Brownian motion with positive drift starting at 0. The process with reflection at fixed barrier $b<0$ (sometimes called a "regulated Brownian motion") is: \begin{equation} \...
Dale123's user avatar
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7 votes
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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 $\sigma(b)=0$. ...
ColorfulLion's user avatar
1 vote
1 answer
78 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 ...
user1598's user avatar
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2 answers
86 views

Find the distribution of maximum of $B_t-t$

Let $B_t$ be a standard Brownian motion. It is easy to show that $\sup B_t-t<\infty$ a.s. . The question is, can we determinate the distribution of $\sup_{t\in [0,\infty)}B_t-t$?
Tiancheng's user avatar
4 votes
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122 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)...
Timothy Budd's user avatar
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3 votes
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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}\...
David Geldbach's user avatar
3 votes
1 answer
141 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 ...
Mueller's user avatar
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Conditioned random walk over a graph

I want to solve for a conditioned random walk over a graph. I have a directed graph $G$. The random walkers start at a fixed node, Source. They all need to end up at fixed node, Sink. So the random ...
highBandWidth's user avatar
5 votes
1 answer
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On the convergence of a martingale

Let $W$ be a standard one dimensional Brownian motion and let $A$ be the process defined by : $$\forall \ t\geq 0: \quad A_t := \int_0^t\left(1 + e^{W_s}\right)\mathrm{d}s$$ and for $t\geq 0$, we ...
Greyearl's user avatar
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1 answer
123 views

Polar form of 2D Brownian motion

Consider two independent unidimensional Brownian motion $w_1$ and $w_2$. What is the polar form of $(w_1,w_2)$? If $r(t)$ and $\phi(t)$ satisfy $(w_1,w_2) = r(t)(\cos(\phi(t)),\sin(\phi(t)))$, how to ...
happy hello's user avatar
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1 answer
179 views

Macroscopic sets - a notion of largeness for Lebesgue null sets

Let $E$ be a measurable subset of $\mathbb R$. We say $E$ is $\alpha$-macroscopic, for $0 \leq \alpha \leq 1$, if there exists an $\alpha$-Holder continuous function $f: \mathbb R \to \mathbb R$ such ...
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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 "...
Andrea Marino's user avatar
4 votes
1 answer
100 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 ...
Focus's user avatar
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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 ...
Benjamin's user avatar
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2 votes
2 answers
198 views

Weak convergence of measures on continuous function spaces

Let $S$ be the unit sphere of $C[0,1], \|\cdot\|_{\infty})$, let $(B_{t})_{t}$ the brownian motion. I would like to show that the measure $\mu_r$ defined on $\mathbb{B}(S)$ by $\mu_r(A):=P\Big(\frac{...
Paul's user avatar
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2 answers
315 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 ...
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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 ...
arrhhh's user avatar
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2 votes
1 answer
118 views

Measurability of two hitting times at the stopped $\sigma$-algebra

Let $\mathcal{F}=(\mathcal{F}_t)_{t\ge 0}$ be the complete filtration generated by the Brownian motion $B $, and let $a<0<b$. Define the stopping times $\tau_a=\inf\{t\ge 0\mid B_t=a\}$ and $\...
John's user avatar
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0 answers
139 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 ...
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0 answers
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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 ...
sara's user avatar
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2 votes
0 answers
60 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 ...
Thomas Kojar's user avatar
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2 votes
1 answer
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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(...
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1 answer
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Resources to understand Lebesgue measure of Brownian motion's path [closed]

[https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Hansen.pdf][page 12] and [peter morters][page 47] Let $B$ be a stanrd Brownian Motion and $R$ a function defined on $\mathbb{R}^2$ such ...
sara's user avatar
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0 answers
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Application of Ito's formula to Liouville's theorem

Liouville's theorem for bounded harmonic functions could be proved using Ito's formula, martingale convergence and Blumenthal's 0-1 law. I tried checking the classical books on Brownian motion and ...
mathex's user avatar
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2 votes
1 answer
133 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 $\tau_D = \inf\{t > 0 : B_t \notin D\}$...
Focus's user avatar
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3 votes
1 answer
280 views

Each diffusion SDE is associated to a *unique* family of transition kernels

I consider an SDE of the form $dX_t=b(X_t) \, dt + \sigma(X_t) \, dW_t$, with $b$ and $\sigma$ globally Lipschitz on $\mathbb{R}^n$. How can I prove that there exists a unique family of transition ...
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1 vote
0 answers
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Locality and restriction properties for self-avoiding and loop-erasing random walks

This question has been cross-posted from math.stackexchange.com : https://math.stackexchange.com/questions/4742746/locality-and-restriction-properties-for-self-avoiding-and-loop-erasing-random-wa I ...
Testcase's user avatar
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1 vote
1 answer
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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 ...
Benjamin's user avatar
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3 votes
1 answer
140 views

Recurrence of Drifted Brownian Motion Conditioned to not hit Moving Barrier

Suppose we have a Brownian motion $X$ with $X_0>0$ and drift $\mu$ conditioned to be less than a barrier $R$ which has behaviour $R_0 = r$, $dR_s = \nu \, ds$, where $\mu > \nu > 0$. Can we ...
user1598's user avatar
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1 vote
1 answer
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Volterra Processes (integration wrt Brownian motion): reference request

I need some references about Volterra processes $Y=(Y_t)_{t\geq0}$ defined as $$ Y_t:=\int_{0}^{t} g(t,s)dB_s, \ \ t\geq 0,$$ where $B=\left(B_t\right)_{t\geq0}$ is a brownian motion and $g$ satisfies ...
Joegin 's user avatar
4 votes
0 answers
183 views

Schrödinger Bridge for other costs

Stochastic control formulations of the Schrödinger bridge problem between $\mu,\nu$ are well known (e.g Chen et al Eq. 4.23) $$\inf \limits_{p_t, v_t} \int_0^T \int \frac{1}{2}\lvert v_t\rvert^2 p_t ...
nico's user avatar
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0 answers
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Calculation of the difference of two Brownian bridges

I was told that the difference of two independent brownian bridge process is $\sqrt{2}$ times a brownian bridge process, i.e., $$B_{1t} - B_{2t} = \sqrt{2}B_t$$ where $B_{1t}$ and $B_{2t}$ are ...
John Smith's user avatar
7 votes
2 answers
380 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 $...
El_mago's user avatar
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2 votes
1 answer
130 views

Local martingale with increasing process

Here is a problem in stochastic calculus: If $M_t$ is a continuous process and $A$ an increasing process, then $M$ is a local martingale with increasing process $A$ if and only if, for every $f\in C^2$...
Liu Wei's user avatar
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0 answers
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Expand White Noise and Brownian Motion in Haar basis: which version of Haar basis?

Start with the Haar basis of $L^2(\mathbb{R})$, namely, the functions $$ \chi(t-k) \text { and } 2^{j / 2} h\left(2^j t-k\right), j \geq 0, k \in \mathbb{Z}, \quad \quad \quad (1) $$ where $\chi(t)$ ...
Mark's user avatar
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2 votes
0 answers
266 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:...
Fawen90's user avatar
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2 votes
1 answer
233 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,...
leo monsaingeon's user avatar
5 votes
3 answers
894 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 ...
Mr H's user avatar
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1 vote
0 answers
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Probability that a Lévy process "closely" follows a predefined trajectory

For a Brownian motion $(B_t)_{t\geq 0}$ it is well-known [Thm 38, David Freedman, Brownian motion and diffusion], that if $f:[0,1] \to \Bbb R$ is a continuous function with $f(0)=0$ then for $\...
Falrach's user avatar
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1 vote
2 answers
184 views

Converse Cameron-Martin theorem for shifts by adapted processes

Let $W$ be a standard one dimensional Brownian motion, $\mathcal F_t$ its natural filtration, and $\mathbb P$ be the induced Wiener measure on $\Omega := C[0, 1]$. Given a $C[0, 1] $ valued random ...
Nate River's user avatar
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4 votes
1 answer
342 views

Derive the solution of the diffusion equation from the solution of a random walk

Summary The probability distribution (pdf) of a random walk in 1 dimension is represented by a Bessel function. On the other hand, the pdf of a Brownian motion in free space is represented by a ...
Sam's user avatar
  • 261
1 vote
1 answer
81 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 ...
ssss nnnn's user avatar
2 votes
0 answers
158 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 $\mathbb{R}^2$ ...
ssss nnnn's user avatar
8 votes
2 answers
379 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 $B_t\in C^s(x)$. ...
pipenauss's user avatar
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0 votes
1 answer
157 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 ...
PhD_InStochastics's user avatar
5 votes
2 answers
625 views

Brownian bridges as conditioning

Brownian bridges are interpreted as Brownian motions conditioned to start and end at given points. However, I have not seen a source that makes this precise, though this may be due to my own lack of ...
Nate River's user avatar
  • 4,802
1 vote
1 answer
273 views

SDE with non-degenerate diffusion visits every point

I am asking an extension of the question here for SDEs of the Ito form. Consider the SDE $dX_t =\sigma(X_t) dW_t$, where $W$ is a $d$-dimensional Brownian motion and $\sigma:\mathbb{R}^n\to \mathbb{R}...
John's user avatar
  • 483
2 votes
2 answers
120 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 ...
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