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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: Problem with some definition.

Let $(B_t)$ a Brownian motion. Let $f:\mathbb R\to\mathbb R$ such that $f\in\mathcal C^2(\mathbb R)$ and such that $f''$ is bounded. Show that $$\lim_{h\to 0^+}\frac{\mathbb E[f(B_{t+h})\mid B_t=x]-f(x)}{h}=\frac{f''(x)}{2}.$$ In the proof, it's…
idm
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Brownian motion under Girsanov change of measure

i am struggeling with the following exercise Let $r,\mu,\sigma,T>0$ and consider the market model with a money-market account $B$ and one risky asset $S$ such that \begin{eqnarray*} dB(t)&=&rB(t)dt,\quad B(0)=1,\\ dS(t)&=&S(t)(\mu dt+\sigma…
Seneca
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Brownian motion, why $p\{B_{T_{x+h,x-h}}=x\pm h\}=\frac{1}{2}$?

Let $(B_t)$ a Brownien motion. Let $\tau_a=\inf\{t\geq 0\mid B_t=a\}$ (with $a\neq 0$) and $T_{a,b}=\tau_a\wedge \tau_b$. Suppose $x\in[a,b]$ and $B_0=x$. Let $h>0$ very small. Why do me have $p\{B_{T_{x+h,x-h}}=x\pm h\}=\frac{1}{2}$ ? The argument…
idm
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Reflected brownian motion at arbitrary lower barrier

What is the kernel for reflected Brownian motion at some lower barrier $p_b$? The best I could come up with is: $(e^{-((x-u)/a)^2/2}+e^{-((x+u-2b)/a)^2/2)})/(2\pi)^{1/2}$ Which is equal to zero when the derivative is taken and $x=p_b$ But this…
Thomas Baert
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Why is Brownian Motion B_t distributed as N(0,t)?

Almost all textbooks define a Brownian Motion ($B_t$)using three / four points: $B_0 = 0$; it has stationary independent increments; for every $t>0$, $B_t$ has a normal $N(0,t)$ distribution; it has continuous sample paths. Could someone please…
John Smith
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Probability of Position of Brownian motion at hitting time

this might be a stupid question but I am a bit stuck here. let $B$ be a standard Brownian motion and $H_a$ the first hitting time of level $a$. I now want to find the probability $\mathbb{P}(B_{H_a} \in dw | H_a \in dt)$ Of course I know that…
Saali
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given SDE how to find martingale measure

I've been stuck with the question how to find a measure to make a discounted price a martingale. I cannot use Girsanov because I am only given the SDE for which an unique strong solution exists but not known. So, I have $dX_t = (\frac{1}{X_t} -…
Saali
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Let B(·) and W (·) are two independent Brownian motions. Show two integrals have the same distributions.

Let B(·) and W (·) are two independent Brownian motions. How to show that the distributions of $\int_{0}^{1}(B(t)+W(1-t))^2dt$ and $\int_{0}^{1}((B(t))^2+(B(1)-B(t))^2)dt$ are the same? I think that it suffices to show their characteristic functions…
user114757
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Local maxima and minima of Brownian motion

I have trouble understanding why the Brownian motion is nowhere differentiable, and I found somewhere that after showing the total variation of the Brownian motion is $+\infty$, the author claims that "as a result (of the infinite total variation),…
amazonfire
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Meaning of Brownian motion to distribute as Normal distribution

Let $B(x)$ be a Brownian motion. We know by definition that $B(x+1) - B(x)$ ~ $N(0,1)$. I am trying to understand if that means that $B(x+1)-B(x) = PDF(x)$, where PDF(x) is the probability density function of $N(0,1)$ at the point $x$. Can you tell…
Kevinlove
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Continuity of stationary measure of reflected Brownian motion in higher dimensions

For Brownian motion contained in a closed region (specifically a polyhedron), is it true in general that the stationary density on the boundary is continuous with the value on the interior, in a sense similar to "left continuity" in 1D? I think, but…
Lachlan
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Probability of the behavior of a Wiener process : $\mathbb{P}\{W_1>0, W_5 <0\}$

I want to find the following probability : $\mathbb{P}\{W_1>0, W_5 <0\}$ Where $W_t$ is a Wiener process, so it follows the law $\mathcal{N}(0, t)$. My question is : can say that $\{W_1>0\}$ and $\{W_5 <0\}$ are uncorrelated ? In that case…
Gabriel dLN
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Independence of increments of Wiener processes

Let $W_t$ be a Wiener process. In the demonstration provided below (which is right), it is said, knowing $s
Gabriel dLN
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Von Neumann stability analysis for radial diffusion in a sphere

I have to solve a diffusion probleme, radial diffusion into or out of a sphere. So far so good, I have the explicit solution via difference quotient. But I have trouble evaluation the stability analysis. Compared to linear diffusion, where I an…
Marc Laub
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Probability of a Wiener process staying within a wedge

Given $m>0$, what is the probability that a one-dimensional Wiener process $W$ will satisfy $$\left|W_t\right|\leq mt$$ at all times during the period $t \in [0,1]$?
Daniel Franke
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