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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.

A Brownian bridge is a continuous-time stochastic process $B(t)$ whose probability distribution is the conditional probability distribution of a standard Wiener process $W(t)$ (a mathematical model of Brownian motion) subject to the condition (when standardized) that $W(T) = 0$, so that the process is pinned at the origin at both $t = 0$ and $t = T$.

44 questions
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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{P}(\forall t\in[0,T],W_t\geqslant…
Tuvasbien
  • 9,698
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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 following facts as an exercise: $X$ is a…
Pedro Gomes
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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 Brownian motion conditioned to hit 0 at time T,…
Arthur Boivert
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Draw successively from a deck. Score = no. of black cards minus no. of red cards. Optimal strategy convergence?

During an interview, I was asked the following problem: We shuffle a deck of 52 cards (26 red and 26 black cards), and enter the following the game: starting with a score of $0$, we will draw a card from the deck. If it is red, we decrement our…
Noomkwah
  • 618
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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 $0$ otherwise) as $\frac{2}{\pi}\arcsin(\sqrt{x}),…
Juan Carlos Ortiz
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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 proof (Lemma 15 and Lemma 16), they are using that…
Dovahkiin
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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 the Brownian motion provide a simple upper bound…
Michael Li
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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 $\Pr(\sup_{t \in T} |x(t) + y(t)| > z) \ge \Pr(\sup_{t \in T} |x(t)| > z)$ for all $z > 0$?
Rob
  • 441
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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; c) replacing the opposite color. For each red…
Aguazz
  • 165
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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
mnmn1993
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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) $\mathcal{G}_t$? For…
Bravo
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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 $(B_t)_{t\in[0,1]}$ is $B_t=(1-t)W_{\frac{t}{1-t}}$ where…
Tuvasbien
  • 9,698
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1 answer

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
nullUser
  • 28,703
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Brownian Bridge is not adapted to the natural filtration of the underlying Brownian Motion

Let $W=\{W(t)\}_{t \in [0,\infty)}$ be a Brownian Motion and $X=\{X(t):=W(t)-t W(1)\}_{t \in [0,1]}$ the associated Brownian Bridge. I wan't to verify that $X$ is not adapted to the natural filtration of $W$, i.e. not adapted to…
Louis
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Local times and excursions: Justifying the Occupation Time Formula and Deriving the Exponential Law

Let $$ E^+ = \{ e \in E : e(t) \ge 0 \text{ for all } t \ge 0 \} $$ be the set of positive excursions of Brownian motion. For $e \in E^+$ and for $a > 0$, define $$ T_a(e) = \inf\{ t \ge 0 : e(t) = a \} \quad (\text{with the usual convention } \inf…
Random_geometer
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