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Questions tagged [brownian-bridge]

Questions about the Brownian Bridge stochastic process, which is Brownian Motion conditioned to have specific values at two endpoints, most commonly defined as starting and returning to 0, or starting at 0 and arriving at 1.

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Derivation of $E[B_t | B_T]$ for $t<T$

We know that $\begin{bmatrix} B_T \\ B_t \end{bmatrix} $ is a normal Gaussian vector and $\text{Cov}(B_T, B_t) = \min\{T,t\} = t,$ i.e, $$ \begin{bmatrix} B_T \\ B_t \end{bmatrix} \sim \mathcal{N}(\...
Oskar's user avatar
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4 votes
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Levy Arcsine Law for Brownian Bridges?

If $W_t$ is a standard Wiener process, Levy's Arcsine Law gives us the CDF of the random variable $$ \int_0^1 \delta[W_t \ge 0] dt $$ (where $\delta[\text{event}]$ is $1$ if the event happens and $0$ ...
Juan Carlos Ortiz's user avatar
2 votes
1 answer
55 views

Reference request on rough paths sequence of Brownian bridges

I stumbled onto this talk on applications of rough paths to mathematical finance by Professor John Armstrong. On slides 19-25 he presents a fascinating example of a sequence of rough paths that ...
Theo Diamantakis's user avatar
4 votes
1 answer
200 views

Can a Brownian Motion be scaled by a random time still be a brownian motion?

I was reading the proof that if $\tau=\sup\{s\in [0,1]:B_{s}=0\}\wedge 1$. Then $\bigg(\frac{1}{\sqrt{\tau}}W_{\tau t}\bigg)_{t\in [0,1]}$ is a Standard Brownian Bridge in $[0,1]$ from here. In the ...
Dovahkiin's user avatar
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Show that $Z(t) = \left\{ \begin{array}{ll} (1-t)B\left(\frac{t}{1-t}\right) & x \in [0,1) \\ 1 & \, t=1 \\ \end{array} \right. $ is a Brownian bridge

Let $W_t$ be a standard Brownian motion and let $$Z(t) = \left\{ \begin{array}{ll} (1-t)B\left(\frac{t}{1-t}\right) & t \in [0,1) \\ 1 & \, t=1 \\ \end{array} \right. $$ I don't get why $Z$ is ...
Leon's user avatar
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1 answer
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sde for brownian bridge

Let $B(t)$ be a standard Brownian motion and \begin{align*} Y(t)=B(t)-tB(1) && Z(t) = \left\{ \begin{array}{ll} Z(t)=(1-t)B\left(\frac{t}{1-t}\right)& t\in [0,1)\\ 0 & t=1 \\ \end{...
Leon's user avatar
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1 vote
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38 views

Slepian Inequality for Brownian/Gaussian Bridges

I am trying to find any reference related to understanding bounding probabilities of generalized Brownian Bridges. The motivation comes from Slepian's inequality/lemma, which specifically states that ...
Michael Li's user avatar
2 votes
1 answer
112 views

Why is the Ito Isometry used to prove the limit of the Brownian Bridge?

The 1-dimensional equation for a Brownian Bridge, $dY_t=\frac{b-Y_t}{1-t}dt + dB_t$; $0\le t < 1$, $Y_0=a$ has a solution $Y_t=a(1-t)+bt+(1-t)\int_0^t\frac{dBs}{1-s}$; $0 \le t < 1$. Solutions ...
user86422's user avatar
5 votes
1 answer
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Proving $ X_t = W_t - \int_0^t \frac{X_s}{1 - s} \, ds$ is the Brownian Bridge

Brownian bridge. Let $B$ be a $d$-dimensional Euclidean Brownian motion. Then the process $t \mapsto X_t = B_t - tB_1$ is called a Brownian bridge. Let $G_t = \sigma \{B_s, s \leq t; B_1\}$. Prove the ...
Pedro Gomes's user avatar
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Distribution of the maximum of a rescaled Brownian bridge?

Let $B_t$ denote a standard Brownian bridge, $t \in [0, 1]$. I am interested in understanding the probability $$ f(\lambda) = P\Big(\sup_{0 \leq t \leq 1} \frac{|B_t|}{\sqrt{t(1-t)}} > \lambda\Big),...
Drew Brady's user avatar
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4 votes
1 answer
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Is a Brownian Bridge less dispersed than a Brownian Motion?

Consider the standard Wiener Process/Brownian Motion $W(t)$ on $[0,1]$ and the corresponding Brownian Bridge $B(t)=W(t)-\frac{t}{T}W(T)$. I am interested to know if the boundary crossing results for ...
Michael Li's user avatar
2 votes
1 answer
184 views

PDF of the Brownian Bridge

I am self-learning introductory stochastic calculus from the text, A first course in stochastic calculus, by L.P.Arguin. Exercise 2.6 asks to find the PDF of a brownian bridge. We have: Definition (...
Quasar's user avatar
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Show that a process is a local martingale (Brownian bridge)

Let $W=\{W_t\}_{t\geq 0}$ be a Brownian motion and $\{X_s\}_{t\leq s\leq1}$ be a Brownian bridge. Let we have a value function $V^*:[0,1)\times\mathbb{R}\cup\{(0,1)\}\rightarrow \mathbb{R}$ given by \...
what_456's user avatar
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Itô Integral and Brownian Bridge Process

I understand for some function $k: [0, 1] \to [0, c]$, the Itô integral is $$\int_{0}^{1} k(r) \, \mathrm{d}B(r) = \lim_{n\to\infty} \sum_{i=1}^{n} k(r_{i-1}) [B(r_i) - B(r_{i-1})] $$ for a standard ...
kpr62's user avatar
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$(1-t)\int_0^t \frac{1}{1-s}\,dB_s\overset{d}{=} B_t-tB_1$?

Let $0<t<1$. I want to show that $$ (1-t)\int_0^t\frac{1}{1-s}\,dB_s\overset{d}{=}B_t-tB_1. $$ I tried using Ito's formula for a standard process. That is for $f(t,x):=(1-t)x$, denoting $X_t:=\...
mathmd's user avatar
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Does the distribution of the maximum increase when adding independent Gaussian processes?

Let $x(t)$ and $y(t)$ be independent, mean-zero Gaussian processes, indexed over some general metric space $T$. Is it true that $\Pr(\sup_{t \in T} |x(t) + y(t)| > z) \ge \Pr(\sup_{t \in T} |x(t)| &...
Rob's user avatar
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1 answer
113 views

Conditional distributions of points on Brownian Bridge with Gates

Suppose I have a standard Brownian bridge $B(t)=(W(t)|W(0)=W(1)=0)$. Suppose further there is a finite set $S \subset [0,1]$ where we denote $s$ a generic element of $S$. Each $s$ is associated with a ...
johaschn's user avatar
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1 vote
1 answer
131 views

The covariance of a multivariate Empirical Process

Van der Vaart states in his book "Asymptotic Statistics" that an empirical process (of the sample $X_1, X_2, \dots, X_n$) is defined by $$G_n = \sqrt n( P_n - P),$$ where $P$ is the true ...
lmaosome's user avatar
  • 632
2 votes
0 answers
86 views

Computing the expected quadratic variation of a Brownian bridge

Consider the Brownian bridge process $$B_t = W_t - tW_1,$$ where $W_t$ be a Brownian motion on $[0,1]$. What is the expected quadratic variation of $B_t$? Definition: The co-variation of two ...
Quertiopler's user avatar
1 vote
0 answers
44 views

Integrating a Brownian bridge with respect to a deterministic function

I am faced with the task of calculating the distribution of the following integral: $$I(\omega) := \int_0^1 B_{G_-(t)}(\omega)\,\mathrm dF(t),$$ where $(B_t)_{t\in[0,1]}$ is a Brownian bridge, i.e. a ...
Syd Amerikaner's user avatar
5 votes
1 answer
219 views

Conditional expectations and Brownian bridge

I'm currently working about Brownian Bridge and I have to compute the following expectation \begin{equation} E[W_tW_s\vert W_T]\end{equation} We consider here that the Brownian bridge is defined as a ...
Arthur Boivert's user avatar
0 votes
1 answer
35 views

Finding the expectation of a characteristic function conditioned on a gaussian random variable.

For $X_t$ is a Brownian bridge, I have to find $E[e^{iu(X_{4/5}-\frac{1}{2} X_{3/5})}|X_{3/5}]$. I can find the distribution of $X_{4/5}-\frac{1}{2} X_{3/5}$ with no issues, and I can see that $E[e^{...
Risuka's user avatar
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2 answers
89 views

Finding the probability of an increment of a Brownian Bridge

I've been given a process $X_t=(1-t)B(\frac{t}{1-t}),0\leq t\leq 1$ where $B(\frac{t}{1-t})$ is a standard Brownian Motion. So far, I have proven this is a Brownian bridge. Now I need to find $P(X_{4/...
Risuka's user avatar
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2 votes
1 answer
155 views

Is the average of Brownian bridge Gaussian?

Let $\{W_t\}$ be standard Brownian motion. Let $B_t=W_t-tW_1$ be Brownian bridge on $[0,1]$. Let $\mu$ be a Borel probability measure on $[0,1]$. I want to show that $\int_0^1 B_t\mu(dt)$ is a ...
A. Howells's user avatar
3 votes
0 answers
144 views

Ornstein-Uhlenbeck Bridge as a Random Walk Limit (The Urn Game)

An urn contains $N$ red balls and $N$ black balls. Consider the game in which you sequentially draw balls from the urn: a) with replacement; b) without replacement, until the $2N$ balls are all drawn; ...
Aguazz's user avatar
  • 143
3 votes
0 answers
188 views

Density function of a Brownian motion conditioning on a first exit time

Let $B_t$ be a one dimensional Brownian motion such that $B_0=a$. Here $t \in [0,T]$. Define the first exit time $\tau : =\inf\{ s \in [0,T] : B_s=b\}$ with $b<a$. I would like to find the ...
mnmn1993's user avatar
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0 votes
0 answers
75 views

Conditional probability of Brownian bridge

Suppose $B_{0,a}^{T,b}(t)$ is a Brownian bridge such that $B_{0,a}^{T,b}(0)=a$ and $B_{0,a}^{T,b}(T) = b$. The probability density function of $B_{0,a}^{T,b}(t)$ is the conditional probability density ...
mnmn1993's user avatar
  • 395
2 votes
1 answer
127 views

How to prove a binary decomposition of $x(1-x)$?

Trying to find the variance of Brownian bridge (maybe not in the standard way) I settled on the formula: $x(1-x) = \sum\limits_{k=1}^{+\infty}2^{k-1}\left(-\frac{b_k}{2^k}+\sum\limits_{i=k+1}^{+\infty}...
Andrey Rogatkin's user avatar
2 votes
0 answers
119 views

A generalization of the KMT theorem for empirical processes

Recall the KMT embedding for empirical processes: For all $n\geq 1$, there exists a probability space with $(U_k)_{1\leq k\leq n}$ i.i.d with uniform $[0, 1]$ distribution and $W_0$ a Brownian bridge ...
Papagon's user avatar
  • 315
1 vote
0 answers
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Level reaching of a Brownian bridge

Let $W$ denotes a Wiener process and $\gamma=\max\left\{ t\in\left[0,2\right]:W_{t}=0\right\}$ . What is the probability, that $$\mathbf{P}\left(\max_{s\in\left[0,\gamma\right]}W_{s}>\sqrt{\gamma}\...
Kapes Mate's user avatar
  • 1,424
1 vote
1 answer
171 views

How to prove $W_t-tW_1$ is Markov?

For a Brownian motion $W_t$, how do we prove the bridge process $W_t-tW_1$ is Markov? Essentially, we need to prove for $s< t$, \begin{align} \mathbb{P}(W_t-tW_1\in x\mid \mathcal{F}_s)=\mathbb{P}(...
Bravo's user avatar
  • 4,453
0 votes
1 answer
294 views

Proving that a process is Brownian Bridge

This is from Durrett's Probability : Theory and Examples , excercies 8.4.2. $B_t$ is brownian motion starting from $0$. Show that $X_t = (1-t)B(\frac{t}{1-t})$ is Brownian Bridge. I managed to show ...
Dongri's user avatar
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3 votes
1 answer
588 views

Is the Brownian bridge a martingale with respect to its own filtration?

Define $B_t=W_t - tW_1$. Let $\mathcal{F}_t$ and $\mathcal{G}_t$ be the filtrations generated by $\{W_t\}$ and $\{B_t\}$ respectively. Is $B_t$ a martingale wrt (i) $\mathcal{F}_t$ and (ii) $\mathcal{...
Bravo's user avatar
  • 4,453
3 votes
0 answers
130 views

Hitting-time Brownian Bridge

Let $(B_t)_{t\in[0,1]}$ be a standard Brownian bridge, that is $B_0=B_1=0$. I'm interested in $\tau_a:=\inf\{t\in[0,1],B_t=a\}\in[0,1]\cup\{+\infty\}$ for $a>0$. One of the choices for $(B_t)_{t\in[...
Tuvasbien's user avatar
  • 9,282
-2 votes
1 answer
224 views

conditional expectation and variance of integral stochastic of a Geometric Brownian motion

Let $\sigma_s$ be a Geometric Brownian Motion (GBM), ie, $$ \sigma_s =\sigma_0 \exp(\frac{-\alpha^2}{2}s+\alpha W_s) $$ where $W_s$ is a Standard Brownian Motion. Calculate the conditional Expectation ...
JC25's user avatar
  • 3
11 votes
0 answers
352 views

Probability that a $d$-dimensional Brownian bridge is greater than a given parameter

Let $(W_t)_{t\in[0,T]}$ be a Brownian bridge such that $W_0=a$ and $W_T=b$, the probability that $\forall t\in[0,T],W_t\geqslant x$ given the parameter $x\leqslant\min(a,b)$ is well known : $$ \mathbb{...
Tuvasbien's user avatar
  • 9,282
3 votes
1 answer
466 views

Intuition for formula of variance of Brownian Motion conditioned on two endpoints

I am familiar with the following interpolation property of Brownian Motion (this is essentially a theorem about a Brownian Bridge): Theorem. Let $W$ be a standard Brownian Motion, and $0<t_1 < t ...
nullUser's user avatar
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