# Questions tagged [brownian-motion]

The brownian-motion tag has no usage guidance.

371
questions

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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
...

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### 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 ...

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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 ...

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### 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 $...

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### 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 $f\in C^2$...

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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 $\chi(t)$ ...

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### 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:...

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### 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,...

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### "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 ...

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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 $\...

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120
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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $\mathbb{R}^2$ ...

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### 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 $B_t\in C^s(x)$.
...

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### Limit of non-increasing sequence of stopping times

I am trying to prove the next proposition:
Let $\sigma$ be a stopping time and $W$ be a Brownian motion. Consider the set
\begin{equation}
A=\{\tau\, |\,\tau\leq \sigma, \,W_\tau\sim W_\sigma\}
\end{...

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### 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 ...

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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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130
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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\})=\...

2
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215
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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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78
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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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176
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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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369
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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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113
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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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327
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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 ...

3
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292
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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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### Almost sure equivalence of regular conditonal probability of a progressive process in a controlled SDE equation

I already asked this question in MSE (see https://math.stackexchange.com/questions/4527325/almost-sure-equivalence-of-regular-conditonal-probability-of-a-progressive-proce).
Consider the (canonical ...

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0
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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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169
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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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### A two-dimensional variant of Bessel stochastic differential equation

Let $Z$ be a complex Brownian motion starting at $0$. The stochastic integral
$$W = \int_0^t \frac{Z_s}{|Z_s|} \mathrm{d}Z_s.$$
yields a complex Brownian motion (starting at $0$). The natural ...

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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. ...

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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 ...

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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}\...