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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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Brownian motion hitting time is finite

Let $\{B_t:t\geq0\}$ be a $d$-dimensional Brownian motion. $D\in\mathbb{R}^d$ is a bouded open set, $\sigma:=\inf\{t\geq0:B_t\in\partial D\}$. I want to prove that $\mathrm{P}_x(\sigma<\infty)=1$. My proof is correct? To prove this, it is enough to…
sate
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A 0-1 law of Brownian motion hitting time

Define the first hitting time $\tau^x_A:=\inf\{t>0, B^x\in A\}$, B is a standard Brownian motion, my question is: if $P(\tau_A^x<\infty)>0$, then $P(\tau_A^x<\infty)=1$? For 1, 2 dimensional Brownian motion, it is true. But what about dimension…
Luffy
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What is the average distance in 1D Brownian motion?

I would like to understand the time-dependence of the average distance from the start-point in 1D Brownian motion. It's said the expected distance in Brownian motion is 0, which I would call the average end-position, including (-) signs. But here I…
KaPy3141
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Brownian motion: How would a philosophical alternative look like?

Without being educated in high Mathematics, I randomly claim that the relation of expected distance to time (spread = time^(0.5)) is based on the probability of a random direction to lead to an increase of distance. This probability is 50%! Now,…
KaPy3141
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Problem to understand the proof of the reflexion principle of Brownian motion in wikipedia

We want to prove that $$\mathbb P(\sup_{0\leq s\leq t}W_s\geq a)=2\mathbb P(W_t\geq a).$$ Here is the proof. My mistakes are in the proof of $$\mathbb P(\sup_{0\leq s\leq t}W_s\geq a, W_t
John
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Levy modulus of continuity for a martingale

Given a Brownian motion $B(t)$ then (Levy, 1937) \begin{equation} \mathbf{P}\bigg(\lim_{h\rightarrow 0}\frac{\sup_{0\le t\le 1-h}|B(t+h)-B(t)|}{ \sqrt{2hlog(1/h)}}=1\bigg)=1 \end{equation} Can the result (or a similar result) still hold for a…
DrM
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Position of 2D Brownian motion exiting quarter plane

Let $X_t = (X_t^1,X_t^2)$ a planar brownian motion without drift with independent components startet at $X_0 = (1,1)$ and $\tau := \inf \lbrace t\ge 0: X_t \notin (0,\infty)^2 \rbrace$ the first time the process leaves the positive quadrant. So one…
maliesen
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Convergence in $L^{2}(\Omega)$

Let $T>0$ and $P^{n}:=\lbrace0=t_{0}^{n}
czachur
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Is the following a Wiener process?

This is a worked example on Wiener processes. Question: Pick a normally distributed random variable $Z \sim N(0,1)$, then define $W(t) = Z\sqrt{t}$. Is $W(t)$ a Wiener process? Answer: It is continuous. $W(0) = 0$. Therefore two required…
user1905552
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Simple Bayes law on Brownian motion

I have seen this identity in some economics journals, but cannot derive it myself. The identity is formulated as follows: Let's say the correct realization of $\mu$ is either 1/2 or -1/2. And the prior on the correct realization is 1/2. Let $X_t$…
Sung Woo Hwang
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Limiting behavior of Standard Brownian motion

Suppose $X(t), t \ge 0$ is a standard Brownian motion. I need to compute $\lim\limits_{n \longrightarrow \infty} \frac{X(n)}{n}$, where, $n = 1, 2, 3, \ldots$. Since $X(t), t \ge 0$ is Gaussian, intuitively I can see that the when $n$ becomes very…
user2348674
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Brownian motion question

Let $B(t)$ denote the standard Brownian motion and let $X(t)$ denote a Brownian motion with $X(0)=0$, drift $0$ and variance $9$. Find the distribution of $aB(s)+bB(t)$, where $a,b,s,t$ are real numbers and $0
Alvin
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Geometric Brownian Motion Process Problem

I would like to ask a question related to Geometric Brownian Motion again. Thanks in advance. Question: Suppose that $S(t)$ follows a geometric Brownian Process: dS(t) = μS(t)dt + σS(t)dW(t) What is the process ( that is, $dY(t)$)…
skatip
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(Brownian Motion) Incremental joint probability conditioned on past value

Let $\{X(t), t \geq 0\}$ be a $BM(5,7)$ (Brownian Motion) with $X(0) = 4$. Let $Y = X(5) - X(3)$ and $Z = X(15) - X(10)$. Find the join probability $f_{Y,Z}(y,z)$. I know that $P(Y,Z| X(0) = 4) = P(Y|X(0) = 4)\cdot P(Z|X(0)=4)$ since the…
Almacomet
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