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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

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Brownian Motion as a limit of Simpler Models. Sheldon Ross - Introduction to Mathematical Finance. Expression for probability of positive increments.

I am going through the book - An Introduction to Mathematical finance by Sheldon Ross in which Brownian Motion is expressed as the limit of simpler models. I do not understand how the expression for the probability of positive increments: $p =…
MathMan
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brownian motion continuous property

In Stein and Shakarchi's functional analysis, Brownian motion $B_t$ is defined in terms of a probability space $(\Omega, P)$ with $P$ its probability measure and $\omega$ denoting a typical point in $\Omega$. We suppose that for each $t$, $0 \le t…
MoneyBall
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first passage time for 2 dimensional Brownian motion

Suppose a two dimensional Brownian motion begins with (1,1) and it will stop moving when it hits y-axis. What's the probability of stopping at positive part of y-axis? I only know the reflection principle of first passage times of one dimensional…
user6703592
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Why $\mathbb E[f(B_t)\mid \mathcal F_s]=\mathbb E[f(B_t)\mid B_s]$?

Let $(B_t)$ a Brownian motion and $(\mathcal F_t)$ its natural filtration. In my lecture, to prove that $(B_t)$ has the markov property they do as following : les $f$ measurable and bounded. $$\mathbb E[f(B_t)\mid \mathcal F_s]=\mathbb…
joshua
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If $(B_t)$ is a Brownian motion, then $P(\sup_{t\in [0,\infty )}B_t=0)=P(\sup_{t\in [0,\infty )}B_t<\infty )$.

If $(B_t)$ is a Brownian motion, how can I prove that $$P(\sup_{t\in [0,\infty )}B_t=0)=P(\sup_{t\in [0,\infty )}B_t<\infty )\ \ ?\tag{1}$$ I can't use the fact that $P(\sup_{t\in [0,\infty )}B_t=\infty)=1 $ (because we use $(1)$ to prove this). The…
Bruce
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Stuck on a Textbook Question (Brownian motion)

I'm a beginner in studying Brownian motion with some background in probability theory and I ran into some problems going through the textbook Brownian Motion by Schilling: Problem Setup Let's denote the random position of a particle at time $t$ in…
Jasonli1997
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Brownian motion question regarding iterated law

I am doing an exercise question: define $$ M_{t}=\max \left\{B_{s}: 0 \leq s \leq t\right\}, \quad m_{t}=\min \left\{B_{s}: 0 \leq s \leq t\right\} $$ we are asked to find a number $r$ such that with probability one $$ \limsup _{t \rightarrow…
PaulWH
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Compute $\mathbb P^x\{\sup_{0\leq s\leq t}B_s\geq h, \inf_{0\leq s\leq t}B_s>0\}$ where $(B_t)$ is a Brownian motion.

Let $x\in (0,h)$ and $(B_t)$ a Brownian motion. I would like to compute $$\mathbb P^x\left\{\sup_{0\leq s\leq t}B_s\geq h, \inf_{0\leq s\leq t}B_s>0\right\}.\tag{*}$$ In otherword, I would like to compute the probability that a Brownian starting at…
Bruce
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Geometric Brownian Motion definition

I am a bit confused about how the geometric brownian motion process is commonly defined. On this reference it seems to imply that the $\mu$ and $\sigma$ are the mean and the standard deviation of the normal distribution where the logarithm of the…
mvc
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Does $B_{\tau\wedge t}\sim C\mathbb 1_{[-h,h]}(x)e^{-\frac{x^2}{2t}}$ where $\tau=\inf\{t>0\mid |B_t|\geq h\}$

Let $(B_t)$ a Brownian motion and $$\tau=\inf\{t>0\mid |B_t|\geq h\}.$$ Does $B_{t\wedge \tau}$ has density $$f(x,t)=C\mathbb 1_{[-h,h]}(x)e^{-\frac{-x^2}{2t}}$$ where $C$ is s.t. $\int_{\mathbb R}f_X(x,t)dx=1$ ? Indeed, $(B_{t\wedge \tau})$ is a…
Walace
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$\int_0^tf(B_s)ds=0$ for all $t>0$ implies $f\equiv0$

This question is from Problem 8.2 and its solution (Schilling's "Brownian Motion"). I cannot understand following proof. Problem 8.2 Let $B_t$ be a d-dimensional Brownian motion, $f:\mathbb{R}^d\to\mathbb{R}$ be a continuous function such that…
sate
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$B(T)$ and $B^a(T^a)$ are same distribution? ($T$ and $T^a$ are same distribution)

Let $\{B_1(t):t\geq0\}$ be a one dimensional Brownian motion. Then, $\{B^a_1(t):t\geq0\}$ is also a Brownian motion where $B^a_1(t):=a^{-1}B_1(a^2t)$ for $a\in\mathbb{R}$. Let $\{B_2(t):t\geq0\}$ be a one dimensional Brownian motion independent of…
sate
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Does a Brownian motion depend on a filtration or not?

I found many different definitions of Brownian motion. One is Def 1: $(B_t)$ is a Brownian motion if it's a.s. continuous, $B_0=0$ a.s., has independents increments and $B_t\sim N(0,t)$. An other one is Def 2: $(B_t)$ is a Brownian motion if…
Walace
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Why this is sufficient? (planar brownian motion, convergence in probability)

I have a question about Lemma 7.21 in Le Gall's "Brownian Motion, Martingales, Stochastic Calculus". Let $\{B_t:t\geq0\}$ be a 2-dimensional Brownian motion $(B_1(t),B_2(t))$. $T^a_b:=\inf\{t\geq0:a^{-1}B_1(a^2t)=b\}$ (this have same distribution…
sate
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Expected value of brownian motion for all positive paths

I've got this question but I can't figure it out. Derive the expected value of $B(t_1)$ of all paths that are positive $t_1$ and calculate the expectation for $t_1=1$ and variance$=1$? Thanks
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