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Questions tagged [brownian-motion]

Questions related to Brownian motion, a continuous stochastic process denoted by $W_t$, $t\geq 0$, with independent increments, such that $W(t)-W(s)$ is normally distributed, with $0$ mean and variance $t-s$.

Brownian motion is a continuous stochastic process denoted by $W_t$, $t\geq 0$, with independent increments, such that $W_t-W_s\sim\mathcal{N}(0,t-s)$, i.e. the increments are normally distributed with $0$ mean and variance $t-s$. Links:

Brownian Motion at Wolfram MathWorld

4494 questions
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The probability that a linear Brownian motion will hit a curve

Summary I am trying to estimate the probability that a standard linear Brownian motion will hit some curve. To make things a bit simple, I can assume that the curve is a graph of a function, that is is positive at $t=0$, that it is bounded from left…
Bach
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running maximum of brownian motion and reflected brownian motion

Hi I am learning the theory of Brownian Motion using Morters and Peres' book (http://www.stat.berkeley.edu/~peres/bmbook.pdf). Let $B$ be 1-dim standard Brownian motion and $M(t):=\max_{0\le s\le t} B(s)$. In the book Theorem 2.18 says…
Tim
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Show that $X(t)=t W(1/t)$ is a Brownian motion if $W(t)$ is a Brownian motion.

I am trying to solve a past exam question for which I have its answers. I've got to the end, but the very last and simplest line has confused me. I've spotted some errors and corrected them, but I think that this line is correct and I just don't…
s1047857
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Brownian Bridge Representation

Let $B_t$ be a Wiener Process, then $U_t=B_t-tB_1,~0\le t \le 1$ is a Brownian bridge. Show that $X_t=(1+t)U_{{t}/({1+t})}$ is a Wiener Process. I'm not quite sure how to start this off. Any help would be greatly appreciated.
ActuariallyImpaired
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How to show that $P( \sup_{0 \leq s \leq 1} |B_s| \leq \epsilon) > 0$ for any $\epsilon > 0$?

Let $(B_s)_{s \geq 0}$ be a standard Brownian motion. In order to solve an interesting exercise, I need to show that $P( \sup_{0 \leq s \leq 1} |B_s| \leq \epsilon) > 0$ for any $\epsilon$. I've tried the following general methods, and failed at…
Elle Najt
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Brownian motion proofs using Itos Formula

Using Ito’s formula, write an expression for $\int_0^1(B(s))^2dB(s)$ Not sure exactly if I did this right. Was hoping for feedback. I let $f(x)=x^3$. Then by definition, Itos formula states that $f(B(b))-f(B(0)) = \int_0^bf'(B(s))dB(s) +…
Lebron James
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First hitting time for a brownian motion with two exponential boundaries

I asked a previous related question here: First hitting time for a brownian motion with a exponential boundary Now Let $B_t$ be the standard Brownian Motion. Is the distribution/density of the first hitting time of $B_t$ for two exponential decaying…
gota
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maximum of a brownian motion and its integral

Let $W_{t}$ be a brownian motion and $$ W^{*}_{t} = \max_{s
neticin
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Brownian motion on the circle.

Let $(B_t)$ a Brownian motion on $\mathbb R$. The following puzzled. So, I read for example here that $(e^{iB_t})_{t\geq 0}$ is a Brownian motion on the circle. But using the stereographic projection,…
tiko
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What is the significance of $[t/ \Delta t]$ in Ross' definition of Brownian motion?

Picture is from Ross' Introduction to Probability Models, 11th ed. I understand the definition of $[t/\Delta t]$, I just don't see how it connects to the position at time $t$ (eq. 10.1). For $\Delta t$ small, won't $t/ \Delta t$ be itself much…
goblinb
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Is adaptedness a distributional property?

Suppose $(\Omega,\mathcal F,\mathbb F,\mathbb P)$ is a probability space supporting a Brownian motion $\{W_t\}_{t=0}^T$, where the filtration is the augmented brownian filtration $\{F_t^W\}$ and $\mathcal F=F_T^W$. Let $\{S_t\}_{0 \leq t \leq T}$ be…
Jinglin-Kiki Xu
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local maximum of brownian motion

I have a question: Given two disjoint intervals $[a,b]$ and $[c,d]$, how to prove almost surely we have $$\sup_{t\in[a,b]}B_t\neq\sup_{t\in[c,d]}B_s$$ where $B$ is a standard brownian motion. I have no idea about this problem. Does someone have an…
Higgs88
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Independence of $B(t)-B(s)$ with respect to $\mathcal{F}_s$

Assume that $X,\{Y_\alpha\}$ (where $\alpha \in A$, $A$ can be uncountable) are random variables. If $X$ and $Y_\alpha$ are independent for all $\alpha \in A$, i.e., $\sigma(X)$ and $\sigma(Y_\alpha)$ are independent for all $\alpha \in A$, then…
jpv
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Brownian motion at infinity

This is probably a standard exercise in stochastic calculus but I haven't been able to come up with a proof that relies only on a given set of results. So my question is about proving the following statement. $B$ denotes the standard Brownian motion…
Calculon
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Convergence in $L^2$ and proof of Brownian motion

Could anybody give me some hints on the following question? I was doing some exercises on Brownian motion and found this online: Let $\left \{ X_n \right \}_{n=1}^\infty$ be a sequence of independent $N(0,1)$ distributed random variables, defined…
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