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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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Deriving Ito process with a drift from geometric Brownian motion.

Please help me solve this question. Thank you. Let the Geometric Brownian motion be: $$ \frac{\Delta S}{S} = \mu \Delta t + \sigma \epsilon \sqrt{\Delta t} $$ $\Delta S$ = change in stock price (s) $\mu$ = expected rate of return $\sigma$ =…
Cindy
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proving $X(s+t)|X(s) = x \sim \mathcal{N}(x,\sigma^2 t)$

Let $\{X(t),t\geq 0\}$ be a Brownian motion so that $X(t)\sim \mathcal{N}(0,\sigma^2 t)$. Then for any $s,t>0$, $$X(s+t)|(X(s)= x) \sim \mathcal{N}(x,\sigma^2 t) $$ I get this intuitively (after the time unit $s$, we go another $t$ seconds. We…
Mr. Bromwich I
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Expectation of $X_T^4$ when $X_T$ is log-normally distributed

Let $X_T$ be a random variable with $$\ln(X_T) \sim \mathcal{N}\left(\ln(x)+\frac{T-t}{2},T-t\right).$$ What is $\mathbb{E}\left(X_T^4\right)$?
Yuteng
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Variance of hitting time for Reflected Brownian Motion

Let $B_t$ to be a standard Brownian Motion and $Z_t=|B_t|$ a RBM. Denote $\tau = \inf \{t>0: Z_t=1\}$, how to calculate $Var\ \tau \ $ without/with martingale theory?
Ghost W
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Brownian Motion(symmetry, time reversal and scaling)

How do I prove the symmetry of Brownian motion? ( -w(t) is a Brownian motion?)? Also i read in many places about time reversal and scaling of brownian motions as prepositions. I would like to learn how the proof for the same could be carried out…
Probabilityman
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Expection of Brownian Squared conditional on the end of the path

I have been asked as a brainteaser to compute the value of: $\mathbb{E}[W_t^2|W_T]$ with $t < T$ ? Does anyone know how to proceed ?
BlueTrin
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Known that $(B_t)\;t>0$ is brownian motion , $E(B_2 B_3 B_5) =$?

This is what I tried to do. $$E(B_2 B_3B_5)=E(E[(B_2B_3B_5)|\mathcal{F}_2]))=E(B_2(E[(B_3B_5|\mathcal{F}_2]))\\=E(B_2(E[(B_3-B_2+B_2)(B_5-B_2+B_2)|\mathcal{F}_2])).$$ Please help me !
carla
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Modified Standard Brownian Motion Distribution

I'm studying for an exam and I came across a problem where I'm not sure about a few pieces of the solution. The problem goes as follows: ---start Let $\{B(t), t \ge 0\}$ be a standard Brownian motion. Let$ Y(t)= tB(1/t),t \gt 0, Y(0)=0$. (a) What…
fullofquestions
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Using Ito's Lemma to derive $\int^{T}_0 W_tdW_t =\frac{1}{2} W_{T}^2-\frac{1}{2} T$ where $W_t$ is brownian motion and $W_t=0$

Using Ito's Lemma, derive $$\int^{T}_0 W_tdW_t =\frac{1}{2} W_{T}^2-\frac{1}{2} T$$ where $W_t$ is brownian motion and $W_t=0$ Appreciate a hint ; dont know where to start.
Tiger Blood
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Question about motion.

I don't know how to solve this problem . Please anyone help. Question:- An object start moving from immobility with a acceleration of " f m/s^2". At the end of every t seconds it increases it acceleration by "f m/s^2". Show that after "nt seconds"…
Nimantha
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Show that $(B_t)$ and $(tB_{1_t})$ has the same distribution where ($B_t)_t$ is a brownian motion

Let $(B_t)_{t\geq 0}$ a brownian motion s.t. $B_0=0$. I want to show that $B_t$ and $tB_{1/t}$ has the same law. $$p\{tB_{1/t}\leq x\}\underset{u=1/t}{=}p\{\frac{1}{u}B_u\leq x\}=p\{B_u\leq ux\}$$ but how can $$p\{B_u\leq ux\}=p\{B_u\leq x\}$$ and…
MSE
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Maximum of Brownian Motion and a constant

I am interested in the distribution of $Z(t) = \max\{B(t),m\}$ where $B(t)$ is a standard Brownian motion and $m$ is a constant. By distribution, I mean the distribution of $Z(t)$ for a given $t$. I tried some keywords to do a search but I am not…
marvinthemartian
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What is a general Brownian Motion?

This might be a dumb question, but no textbook ever defines what a "Brownian Motion" is, just what a "Standard Brownian Motion." I always assumed that a Brownian Motion is any random variable that can be represented at $\mu t+\sigma B_t$ where $B_t$…
MathStudent
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What is the variance of a Brownian Motion?

In my attempt to digest the answers to my previous question about stochastic integrals, I have stumbled upon yet another question that I need some clarification on... Simply, what is the variance of a Brownian Motion $W_t$? Is it $t$ or $1$? I ask…
John Trail
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Brownian motion conditional expectation

I need to solve for the following in my model: $E[X_t^i|X_s < K_1, X_t > K_2]$ where $X$ is Brownian motion and $i$ is a real number. any suggestion? I already know about the simpler case: $E[X_t^i|X_s < K_1]$
user81435
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