Questions tagged [brownian-motion]
The brownian-motion tag has no usage guidance.
403
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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 ...
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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}...
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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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Distribution of "occupation times" of Brownian Motion
Let $B_t(\omega)$ be a standard Brownian motion and let $A\in\mathcal{B}(\mathbb R)$ be a Borel set.
I would like to find the distribution of $$Y_A(\omega):=\lambda(\{t\in[0,1]:B_t(\omega)\in A\})=\...
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Relationship between heat kernel and Maxwell-Boltzmann distribution
There appears to be a connection between the heat kernel and Maxwell-Boltzmann distribution, but I have not seen this in the literature before. I'd appreciate any kind comments or corrections/...
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Stochastic integral with non-anticipating integrand
Let $B$ be a Brownian motion. We want to define $$ \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}}}]. $$
To ...
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Upper left Dini derivative of Brownian motion at a hitting time
Let $W$ be a standard Brownian motion. Define the upper left Dini derivative $D^-W$ by
$$D^-W_t := \limsup_{h \to 0^-} \frac{W_{t+h} - W_t}{h}.$$
Fix $a > 0$, and define the stopping time $\tau$ by
...
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Full version of Cameron Martin theorem for Brownian motion
I’m looking for a version of the Cameron Martin theorem for the Brownian motion under random shifts. Here is the precise statement:
Let $\mathbb P$ be Wiener measure on $\Omega := C[0, 1]$. Given a $C[...
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Converse of Itô's formula
Let $f,h,g$ be continuous functions and $B$ a real Brownian motion. We suppose that a.s. $$\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.$$
Prove that $f$ is ...
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Integrated square difference of Brownian bridges
I am doing some work with measuring the distance between distributions, and someone pointed out to me that I should look into calculating the integrated squared difference of two brownian bridge ...
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Quadratic variation of supremum of brownian motion
I would like to know if in some book or how could I compute the quadratic variation of the supremum of the bronian motion $S_t=\sup_{s\in[0,t]}W_s$ where $W$ is a Brownian motion. I was thinking ...
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What is the quadratic variation of $W(B(t))$?
Let $W$ be a two sided real valued Brownian motion. Let $B$ be a one sided Brownian motion independent of $W$. Consider the process $X(t)=W(B(t))$. Is the quadratic variation finite and if it is, what ...
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Forgery theorem: the Brownian motion stays close to any curve with positive probability
In a paper I am reading the authors claim that, if $B$ is a standard BM in $\mathbb{R}$ and $f\in C([0,1],\mathbb{R})$, then for any $\epsilon>0$
$$
\mathbb{P}(\sup_{t\in [0,1]}|B_t-f(t)|<\...
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How does the conditional Wiener measure work?
In the theorem below $P_D$ means the heat kernel in the open $D \subset \mathbb{R}^m$ and $P_m$ is the heat kernel in whole $\mathbb{R}^m.$
I know absolutely nothing about what Brownian bridges are, ...
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Continuity of Wiener measure on open balls
Let $\mu$ be the Wiener measure on $C_0 [0, T]$, the space of continuous functions starting at $0$, under the sup norm.
Question: Is it true that the function $r \mapsto \mu(B_r(x))$ is continuous in $...
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Lebesgue differentiation theorem at a stopping time
Let $W$ be a standard Brownian motion, and $\mathcal F_t$ it’s natural filtration. Let $H$ be a continuous process, adapted to $\mathcal F_t$ and integrable with respect to $W$.
Question: Is it true ...
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Comparison of the numbers of particles surviving forever
Consider two $N\text{-}$particle systems as follows : for $1\le i\le N$,
$$X^i_t=1+\int_0^t(b+\phi^i_s) \, ds+W^i_t \quad\mbox{and} \quad Y^i_t=1+ct+W^i_t,\quad \forall t\ge 0,$$
where $c>b>0$ ...
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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 $0<\beta<\frac{1}{2}$.
...
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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)...
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Is this expectation $\mathbb E\big[{\bf 1}_{\{x+\inf_{0\le t\le 2}W_t>0\}}(W_{\tau}-y)^+\big]$ strictly positive?
Let $(W_t)_{t\ge 0}$ be a standard Brownian motion and $\tau$ be a stopping time lying in $[1,2]$. For $x, y>0$, can we show
$$\mathbb E\big[{\bf 1}_{\{x+\inf_{0\le t\le 2}W_t>0\}}(W_{\tau}-y)^+\...
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Does the convergence of $f_n$ imply the convergence of $\mathbb P[\inf_{0\le s\le t}(W_s-f_n(s))\le 0]$?
Let $(f_n)_{n\ge 1}$ be a sequence of non-decreasing and continuous functions defined on $\mathbb R_+$ and taking values in $[0,1]$. Further, for each $t\ge 0$, $n\mapsto f_n(t)$ is non-decreasing. ...
user478657
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Is this set negligible?
Let $(W_t)_{t\ge 0}$ be a standard Brownian motion starting at zero. Let $f: [0,1]\to\mathbb R$ be a function that is righ-continuous with left limits. Set
$$A:=\left\{\omega\in\Omega: \inf_{0\le t\le ...
user478657
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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 ...
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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 $\mathbb P[\tau=\infty]>0$....
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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 ...
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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 ...
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Estimates on the density of hitting time for planar Brownian motion
Consider a polygon $\Pi \subset \mathbb{R}^2$, and let $T_{\Pi,x}$ be the (random) time a Brownian motion started at a point $x$ in its interior first crosses $\Pi$.
For any such $\Pi$, do there exist ...
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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 ...
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Second Skorokhod embedding in high dimensions
The first Skorokhod embedding theorem says that any random variable $X$ with $\mathbb E X=0$ and $\mathbb E X^2<\infty $ can be written as $X=B_{\tau }$ where $B$ is a Brownian motion and $\tau$ is ...
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Chung's law of the iterated logarithm for Brownian motion
I am looking for a reference that gives a detailed proof of Chung's law of the iterated logarithm for Brownian motion: $$\liminf_{u\to +\infty}\sqrt{\frac{\ln(\ln(u))}{u}}\sup_{r \in [0,u]}|X_r|=\frac{...
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For some $\alpha>0$, $ e^L=P\left(\exp(\alpha\sup_{|s-t|\le\delta}\frac{|B_s-B_t|^2}{|s-t|})<\infty\right) $?
I am reading one lecture note Dynamics for Spherical Models of Spin-Glass and Aging by Alice Guionnet. On page 124, it says that
for some $\alpha>0$,
$$
e^L=P\left(\exp(\alpha\sup_{|s-t|\le\delta}\...
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Quantitative Skorokhod embedding
The Skorokhod embedding theorem says that any random variable $X$ with $\mathbb E X=0$ and $\mathbb E[X^2]<\infty $ can be written as $X=B_{\tau }$ where $B$ is a Brownian motion and $\tau $ is a ...
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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 ...
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Feynman-Kac formula with non-zero boundary condition
Let $D \subseteq \mathbb{R}^m$ be a bounded domain. The Feynman-Kac formula for the heat equation with initial condition $u(t, x) = f(x)$ and boundary condition $u(t, x)|_{\partial D} = 0$ is given by
...
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What new fractional brownian motion (fBm) simulation methods have emerged since 2010? [closed]
I want to describe new methods for simulating fBm, as in the work of Coerjolly and Dieker, but new methods are not very easy to find.
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Alternate proof of Levy’s characterisation of Brownian motion
Levy’s characterisation theorem for Brownian motion states that for a local martingale $X$ with $X_0 = 0$, $X$ is a Brownian motion if and only if it has quadratic variation $\langle X, X \rangle_t = ...
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Does the convergence of drifted Brownian motion imply the convergence of expectation?
Let $(f_{\epsilon})_{\epsilon>0}$ be a family of non-increasing and continuous functions on $\mathbb R_+$ s.t. $f_{\epsilon}(0)=1$ and $f_{\epsilon}(\infty)=0$. Assume that $\epsilon\mapsto f_\...
user420828
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Intersection of Brownian motion and finite variation process
Let $B$ be a standard Brownian motion, and $A$ a process of finite variation on compacts almost surely, not necessarily adapted to the Brownian filtration.
Question: Denoting by $\mathcal L$ the ...
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Why is Branching Brownian Motion log-correlated?
I need some references(or helps) on understanding why BBM is log-correlated. As I understand it, a random field on some metric space $V$ with distance $d$ is log-correlated if $$\mathbb{E}[X_u X_v]\...
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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) \\
...
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Ito formula for fractional BM + drift and supremum bound
Let $W^H$ be a fBm with Hurst parameter $H$ and let $\mathcal{H}$ be its Cameron-Martin space. Then by Girsanov theorem we know that if $\mathbb{P}$ is an fBm measure, it holds that there exists a ...
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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 ...
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The joint distribution of the min and max of a Brownian [closed]
The joint distributions of the brownian and both the minimum and the maximum respectively are known. What could be said about the joint distribution of the maximum and the minimum of a Brownian ...
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Existence of a process on $\mathbb{R}^2$ that looks like two 'independent' brownian bridges $B_1(x)$ and $B_2(x)$ conditioned on $B_1(x)+B_2(x) > 0$
Consider any probability density function $f(x)$ that has mean zero variance one and say all finite moments. You may assume standard normal density if you like.
Given $a_1,a_2>0$, I consider two ...
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First hitting time for a drifted Brownian motion
While the solution for a first hitting time for a drifted Brownian Motion is well known, I want to post a different question.
Take a continuous-time stochastic process $X_t$ and define the the ...
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On the range of Holder continuity of Brownian motion
It is known that Brownian motion is almost surely locally Holder continuous, on a range that is random, i.e. depends on the particular path. This question explores the maximal range on which Brownian ...
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A convergence question in $L^2$ construction of Brownian motion
I feel confused with a particular step in the $L^2$ consturction of Brownian motion.
Let $\{\xi_n \sim N(0,1)\}_{n\geq 1}$ be a sequence of i.i.d Gaussian random variables on some probability space $(\...
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What is the distribution of $2M_1-B_1$ where $M_t$ is the maximum process of the the Brownian motion $B_t$
Let $B_t$ be a standard Brownian motion and let $M_t:=\sup _{s\le t}B_s$ be the maximum process. What is the distribution of $2M_1-B_1$? is it elementary?
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$\lim_{r \to +\infty}\frac{1}{\sqrt{2r \ln(\ln(r))}}(B_r-B_{\left \lfloor{\sqrt{2r \ln(\ln(r))}}\right \rfloor})= 0$ a.s.?
Consider a Brownian motion $B$ and let $f(r)=\sqrt{2r \ln(\ln(r))}.$
Is it true that $\lim_{r \to +\infty}\frac{1}{f(r)}(B_r-B_{\left \lfloor{f(r)}\right \rfloor})= 0$ a.s. ?
If so, how to prove it? ...
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Conditional probability of maximum and minimum of Brownian motion
I want to ask for the following problem. Let $(W_t)_{t\geq 0}$ be the standard Brownian motion. For each $t>0$, we call $$m_t =\inf_{0 \leq s \leq t} W_s, \qquad M_t = \sup_{0 \leq s \leq t} W_s.$...
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