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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Is -W(t) a Wiener process?

Indeed it should be because I have to prove it but I have some stupid I guess trouble. Let $V(t)=-W(t)$ where $W(t)$ is Wiener process. $V(0)=0$ without doubts. let $t>s$ then $V(t)-V(s)=W(s)-W(t) \sim \mathcal{N}(0,s-t)$ but we wanted…
nodis6
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Mixed moment between a Brownian motion in absolute value and another Brownian motion

I have this given problem, how can I calculate this mixed moment? $$E[|B(t)|* B(1-t)]$$ I know that the expected value of the Brownian motion is $0$, so this is a covariance, but the absolute value make me insecure on how calculate the result.
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Reflection principle of Brownian motion does not hold?

How is P{ Bt > a | Ta <= t } = 1/2 an appropriate description of the reflection principle of Brownian motion? Shouldn't the inequality in the denominator be "less than" instead of "less than or equal to"? In the case where Ta = t exactly, I would…
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Can we find the joint density of the triple composed of Brownian motion, its running minimum and its running maximum, Revuz-Yor p. 111, exercise 3.15?

Let $B_t$ a standard real valued Brownian motion, its running minimum $s_t \triangleq \min \limits_{0 \leq s \leq t} B_s$ and running maximum $S_t \triangleq \max \limits_{0 \leq s \leq t} B_s$. From Revuz Yor p.111, we know that for any Borel set…
megaproba
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Expectation of a brownian motion given two states

I have a following question about standard brownian motion. For $0 < s < t < u$. How could I derive $E(B_t|B_s,B_u)$? Here is my try: First, $$E(B_t|B_s,B_u) = E(B_t\cdot B_s|B_u) / P(B_s|B_u)$$ For the numerator, we can calculate that it equals to…
Yu Bai
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Construct Brownian Motion

I got a question about the construction of Brownian motions. Just to be all on the same page, for me a Brownian motion on a certain probability space $(\Omega, \mathcal{A}, \mathbb{P})$ is a stochastic process $(B_t)_{t \in [0,\infty)}$, i.e. each…
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Expected value of product of two Brownian Motions

I have a process $W:= \{W_t\}_{t>0}$ and defined $Y_t = tW_{1/t}$ for all $t>0$ I want to prove that $E[Y_tY_s] = s$ Substituting in the values and extracting the constants out, I am left with $tsE[W_{1/t}W_{1/s}]$ Now for this to be equal to $s$, I…
jinx
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Is narrow escape problem time independent?

I have narrow escape problem here. But for the sake of completeness, it measures the time it takes for a Brownian particle, $\alpha$, escape through a narrow hole, $A$, on an otherwise reflecting surface, enclosing a volume $V$, provided that $a\ll…
ck1987pd
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Prebrownian Motion vs brownian Motion

I am currently reading the book "Brownian Motion, Martingales and Stochastic Calculus" by Jean-Francois Le Gall 2016. In chapter 02, he constructs the brownian motion through the prebrownian motion and the continuitytheorem of kolmogorov. I am…
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Expextation of products of Standard Brownian Motion

$B_{i} $ is a standard Brownian motion. $$E[\prod_{i=1}^3 B_{i}]$$ I know how to find $E[B_{1}B_{2}]$, but how do I find the expectation of this…
CJC .10
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How to show that if we multiply an orthogonal matrix by an n dimensional brownian motion is still a brownian motion

I am currently learning about brownian motion, and I am working on an exercice which it's goal is to show that if we multiply an orthogonal matrix by an n dimensional brownian motion is still a brownian motion. I just wanna know if there is a…
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Is this definition of Brownian motion correct?

I got a definition of BM as Definition 2.3. The Brownian motion is a continuous time stochastic process $\{W(t), t \geq 0\}$ that satisfies the following conditions: (i) $W(0)=0$ a.s.; (ii) the paths $t \longmapsto W(t)$ are continuous a.s.; (iii)…
Akira
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Why $\mathbb P(X_{\tau-t}<0\mid \mathcal F_{\tau})=\frac{1}{2}$?

Let $(B_t)$ a BM (Brownian Motion) and $\tau=\inf\{t>0\mid B_t=a\}$. Set $X_t=B_{t+\tau}-B_\tau$. I know that $(X_t)$ is a BM independent of $\mathcal F_\tau$. In the proof of the reflexion principle on wikipedia, they say that $\mathbb…
joshua
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Compute $\mathbb P(\sup_{t\in [a,b]}B_t>x)$, what's wrong with Markov property here?

I would like to compute $\mathbb P\left(\sup_{t\in [a,b]}B_t>x\right)$ where $(B_t)$ is a Brownian motion and $0x \right)=\mathbb P\left(\sup_{t\in…
joshua
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Let $\{B(t)\,|\,t\geq 0\}$ be standard Brownian Motion. Suppose further $B(8)=0$. Find the probability that $B(4)>1$.

Here is my solution, which I am hoping can either be verified or corrected. $B(t)$ is normal with mean 0 and variance $t$. $P(B(4)>1\;|\;B(8)=0) = P(B(4)+B(8)>1+0\;|\;B(8)=0) = P(B(12)>1) = P(Z>\frac{1}{\sqrt{12}}) = 1-P(Z<0.29) = 1-0.6141 =…
D Clark
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