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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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 ...
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 (...
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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 \...
• 157
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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 ...
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• 431
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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 ...
• 143
1 vote
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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 ...
• 632
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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 ...
• 149
1 vote
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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 ...
219 views

Conditional expectations and Brownian bridge

I'm currently working about Brownian Bridge and I have to compute the following expectation $$E[W_tW_s\vert W_T]$$ We consider here that the Brownian bridge is defined as a ...
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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 ...
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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; ...
• 143
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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 $b<a$. I would like to find the ...
• 395
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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 ...
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