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 $t-s$.

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I want to calculate $\int B(t)^2 dB(t)$ where $B(t)$ is Brownian motion

Let $B(t)$ be Brownian motion. I want to calculate $\int B(t)^2 dB(t)$. definition.A process $\{X(t),0\le t \le T \}$ is called a simple adapted process if there exist times ...
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2answers
150 views

Variance of sum of two ito integrals

I don't really understand how to solve the following problem: Var(X) where X = $\int_0^2 2t dW(t) + \int_4^6 W(t) dW(t)$ If I use $E [(A+B)^2] = E(A^2) + E(B^2) + 2E(AB)$ I get to the point where I ...
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1answer
46 views

I want to show $e^{-\alpha t}B(e^{2\alpha t})$ is a Gaussian process. [closed]

Let $B(t)$ be Brownian motion. Show that $e^{-\alpha t}B(e^{2\alpha t})$ is a Gaussian process. Find its mean and covariance functions. thanks .
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182 views

how to prove $\frac{B(t)+W(t)}{2}$ is a Brownian motion where $B(t)$ and $W(t)$ be two independent Brownian motions.

Let $B(t)$ and $W(t)$ be two independent Brownian motions. Show that $\frac{B(t)+W(t)}{2}$ is also a Brownian motion. Find correlation between $B(t)$ and $X(t)$. thanks for any help
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1answer
99 views

$\mathbb{E} \int_a^b W^3(t)\,dW(t)=?$

Is it true that $\mathbb{E} \int_a^b W^3(t)\,dW(t)=0$, for $a < b \in \mathbb{R}$ I know that for an adapted process $\Delta(t), t\geq 0$, the integral $\int_0^t \Delta(u)dW(u)$ is a ...
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1answer
40 views

Question on Black-Sholes Equation

Consider a call option having a strike price $K$ and exercise time $t$; let $r$ be the nominal rate, $\sigma$ volatility and $S_0$ the underlying asset at $t = 0$. How to show that $C(t, ...
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2answers
93 views

How to show that $\mathbb{E}(\int_0^T t\mathrm \, dW_t) = 0 $?

I just want to know why $\mathbb{E}\left(\int_0^T t \,\mathrm dW_t\right)=0$. I know it's got something to do with the Gaussian distribution but I don't really know what.
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60 views

Limit of stationary increment of Brownian Motion [closed]

Does the following limit $$\lim_{s \to \infty}(B_{t+s}-B_{s})$$ have the same distribution with $B_t$?
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2answers
80 views

Expectation Geometric Brownian Motion

Can someone help show me a simple way to show: $$\mathbb{E}(S_t)= S_0e^{\mu t}$$ for $$ S_t = S_0\exp\left( \left(\mu - \frac{\sigma^2}{2} \right)t + \sigma W_t\right) $$ from this page: ...
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1answer
226 views

Brownian Motion conditional distribution

Let $\{X(u),u\geq0\}$ be a standard Brownian motion. What is the conditional distribution of $X(t)$ given $\{X(t_{1}),\dots,X(t_{n})\}$, where $0<t_{1}<\cdots<t_{n}<t_{n+1}=t$? --So far, ...
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0answers
33 views

Brownian motion minimisation problem

Let $B_t$ be a Brownian motion, let $\sigma > 0$ be fixed and let $X_t$ be a process with fixed beginning value $x_0$ that satisfies $dXt = u_tdt + \sigma X_tdB_t.$ Solve ...
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0answers
52 views

Can anyone explain me this proof about a Brownian Motion?

Prove that the process $W_t=(1+t)U_{t/(1+t)}$ on $[0,\infty)$ is a Brownian motion. $\text{(b)}$ Clearly $Y_0=U_0=0$, and inherits continuity of sample paths from $U_t$ (and hence from $W_t$). ...
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1answer
403 views

Distribution of Brownian Bridge

PROBLEM $U_t = B_t - tB_1$, $B_t$ is a Brownian motion on $[0,1]$. What is a Brownian Bridge and give the twodimensional distributions of the vector $(U_s, U_t)$. I think that a Brownian ...
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426 views

Proof on Brownian Bridge

PROBLEM Let $U_t$ be a Brownian bridge on $[0,1]$ and let $Z$ be a standard normal random variable independent of $U_t$. $(a)$ Prove that the process $W_t = U_t + tZ$ is a brownian motion. $(b)$ ...
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1answer
170 views

Probability Brownian Motion - dependence

Does anyone know how to calculate $P(Z(3)>Z(2), Z(2)>0)$ if $Z(3)$ and $Z(2)$ are on the same sample path, i.e. not independent? I found a solution for the case $P(Z(2)<0, Z(1)<0)$ in ...
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1answer
217 views

Application of Optional Sampling Theorem

Lets assume that Brownian Motion starts from some point $x$ for which $a<x<b$ holds. Let $\tau=\inf\{t:B_t\not\in [a,b]\}$ be a stopping time. Now I want to prove that for $\theta>0$ ,an ...
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1answer
152 views

Two people are looking for each other. Is it faster for both to actively search, or for one to search while the other stays still?

Choose among two actors randomly and place the chosen actor at the origin. Place the other actor in the unit circle uniformly at random. Both actors move at the same speed. Both actors are said to ...
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3answers
850 views

Distribution of stochastic integral

Assume that $\mathrm{d}S = \sigma \, \mathrm{d}W$ with initial level $S(0)$ and where $\mathrm{d}W$ is usual Brownian motion. Now $$A(T) = \frac{1}{T} \int_0^T S(t) \, \mathrm{d}t.$$ ...
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2answers
78 views

how to prove $\int_{0}^{a}B(t)dt\sim N(0,\frac{a^3}{3})$

Let $B(t)$ is Brownian Motion. I want to prove the integral $\int_{0}^{a}B(t)dt$ has normal distribution , $N(0,\frac{a^3}{3})$. means $\int_{0}^{a}B(t)dt\sim N(0,\frac{a^3}{3})$
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1answer
155 views

Proving continuity of Brownian paths

Maybe I'll make myself do a routine exercise by posting it here and then later posting my own solution after someone else posts one. Or maybe not. Maybe it's not as routine as I think it might be. ...
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1answer
132 views

Quadratic Brownian Motion

If $B(t)$ is standard Brownian Motion then can we say that $B^2(t) -t $ is a martingale?? Given the following theorem: If $$ \max_{1<k<n} (t_k-t_{k-1}) \to 0$$ as $n \to \infty$ then ...
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2answers
271 views

Brownian motion is almost surely not differentiable everywhere

Could anyone point out the difference between the statement of the following theorems: 1) For any $t\ge0$, Brownian motion is almost surely not differentiable at t. 2) Almost surely, the sample ...
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1answer
89 views

Brownian motion and stochastic integration

How do I compute the following expectation? W(T) is a standard brownian motion (i.e.) W(T)~N(0,T) $E\left[ W(T)\int _{ 0 }^{ T }{ sdW(s) } \right] $ I know that Brownian motion of disjoint time ...
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2answers
53 views

$E[W_{t/3}W_{t/2} \mid \mathcal{F}_{t/5}]$ where W is a Brownian Motion and $\mathcal{F}$ is the natural filtration?

I am unsure how to go about finding this value. $\mathrm{E}[W(t/3)*W(t/2)|$ $\mathrm{F}(t/5)]$ I assume the trick invovles an additional conditional expectation, but I am not sure how to go about ...
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1answer
79 views

Strong Markov Property Brownian Motion Question

If $\tau$ is a stopping time and $\omega(t)$ is Brownian Motion then the Strong Markov Theorem states that $Z(t)=\omega(t+\tau) -\omega(\tau)$ conditioned on $\{\tau <\infty\}$ is distributed as ...
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2answers
430 views

Is the condition “sample paths are continuous” an appropriate part of the “characterization” of the Wiener process?

Wikipedia has separate articles on "Brownian motion" and "Wiener process" (http://en.wikipedia.org/wiki/Brownian_motion and http://en.wikipedia.org/wiki/Wiener_process ). I am not an expert, but that ...
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33 views

A brownian bridge evaluate at a particular random variable

I was wondering of someone could help with the following. I have a random variable given as $\lambda^{*}=\arg \max_{\lambda \in (0,1)} [B(\lambda)-\lambda B(1)]^{2}/\lambda(1-\lambda)$. I am now ...
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58 views

Upper bounds on the sum in a Martingale process

My question is related the hitting time of not a random walk, but a more general martingale process. Suppose we start with an arbitrary $x_0=x$ with $0\leq x\leq 1$. We compute $x_{t+1}$ from $x_t$ ...
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2answers
2k views

What is the definition of a sample path of Brownian motion?

My question has been asked before at beginner's question about Brownian motion . There was only one answer, which was not accepted. It was probably incorrect, because nothing was said about ...
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1answer
1k views

Correlation coefficient of Wiener process

First, I'm not majoring mathematics. I'm studying economics and during reading a thesis I can't understand the 'wiener process' well. I read some books about it and understand the main idea and ...
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41 views

Probability of winding number in 2D Brownian motion

Let $B_t$ be a 2D Brownian Motion with $B_0 = (1,0)$. Now, express $B_t$ in polars, that is, $B_t = (r(t), \theta(t))$. Let $\tau = \inf\{t > 0 : \theta(t) \geq 2 \pi \}$. What is $\mathbb{P}[\tau ...
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1answer
273 views

Probability of 2D Brownian motion passing through a particular point.

Let $B_t$ be a two-dimensional Brownian motion at time $t \in [0,\infty)$. Fix a point $p \in \mathbb{R}^2$. Is the probability that $B_t = p$ at some $t > 0$ equal to zero? If so, why?
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1answer
191 views

Integral of absolute value of Brownian motion

I know it is a really stupid question and it should be quite easy, but how can I show that $\int_0^{\infty}|B_t|\mathbb{d}t=\infty$ a.s. with $B_t$ being a standard brownian motion? I just don't get ...
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1answer
68 views

question on Brownian Motion stopping time and end state

I came across this equation in my lecture notes, which states: $P(T_a < t , W_t \ge a) = P(W_t \ge a)$ where $T_a = \min\{t \ge 0, W_t \ge a\}$. I'm really confused by this equation: as far as I ...
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2answers
981 views

Expected stopping time of brownian motion

I've been looking at this for some time now and still have no sensible solutions, can somebody help me out please. Say I define the stopping time of a Brownian motion as followed: $$\tau(a) = \min (t ...
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1answer
34 views

Finding a tail-probability with the momentgenerating function

I wonder if it is possible to estimate $\mathbb{P}(X<t)$ with the moment generating function? This question popped up when I tried to proof this estimate $\mathbb{P}(T_a<t)\leq ...
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Conditional covariance

Just a simple question when we have $v_{st} = \operatorname{cov}(B_s, B_t\mid Z)$, where $B_t$ is a brownian motion. I know that the answer is $\min(s, t) - E[B_s Z]E[B_t Z]/E[Z^2]$ but i don't know ...
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1answer
72 views

Brownian Motion hitting random point

I got a problem that seems to be quite standard and easy, but I have lots of problems with it. I do already know that $T_a:=\inf\{t\geq 0: B_t=a\}$ is a stopping time for any $a\in\mathbb{R}$ fixed, ...
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1answer
237 views

hitting time of Brownian motion

I'm desperately trying to prove that for a standard BM $B_t$ the stopping time $T:=\inf\{t\geq0: B_t\geq\sqrt{1+t}\}$ is a.s. finite, i.e. $\mathbb{P}[T<\infty]=1$. I actually tried to play around ...
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27 views

Finding asymptotic behaviour

I have a problem in finding the asymptotic behavior of this sum: $$\sum_{i=0}^{n-1} \bigl|B^2 (t_{i+1})-B^2 (t_i)\bigr|$$ over $[0,T]$ when $h= t_{i+1}-t_i \to 0$ and $B$ is Brownian motion. The ...
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336 views

Exponentials of stochastic processes and Brownian motions

This is my first time looking at problems in stochastic calculus, so please bare with the simplicity of the question. As always, any help is greatly appreciated. 1) Given $X_t=\int_0^ur_sds$ for a ...
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2answers
108 views

can someone explain this notation to me?

$$ dz_t \sim O\left(\sqrt{dt}\,\right) $$ $z$ is a Brownian motion random variable, for reference. I just don't understand what the $\sim O$ part means. I've looked up the page for Big O notation ...
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1answer
112 views

Brownian motion at exponential time

I want to find the law of $B_T$, where $B_t$ is a brownian motion, $T$ is exp-distributed with parameter 1, with $B_t$ and $T$ being independent. My idea is to say that the density of $B_T$ is given ...
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2answers
375 views

Question on the proof of the simple Markov property of a Brownian motion

Today we proofed the (simple) Markov property for the Brownian motion. But I really don't get a crucial step in the proof. The theorem states in particular that for $s\geq0$ fixed, the process ...
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88 views

Bounding the expectation of monotone function of stopping times of Brownian motion

Let $X_t$ be a standard Brownian motion and let $Y_t:=X_t + \epsilon B_t$ where $B_t$ is an independent standard Brownian motion and $\epsilon>0$ is small. Let f be a monotone increasing function. ...
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1answer
471 views

Why is the Brownian motion a multivariate normal distribution?

I have seen in class that for some reasons I forgot, the Brownian Motion has a Multivariate normal distribution, but I am unable to prove it easily. Could someone tell me why it's true? From what I ...
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43 views

First hitting time for a brownian motion with two exponential boundaries

I asked a previous related question here: First hitting time for a brownian motion with a exponential boundary Now Let $B_t$ be the standard Brownian Motion. Is the distribution/density of the first ...
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2answers
1k views

Variance of stochastic integral of brownian motion

How do i compute this integral? $ Var [\int_0^T W(t)dW(t)] $ I know the following $E [\int_0^T W(t)dW(t)]$ is 0 but i'm not sure how to apporch the above
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1answer
118 views

Brownian bridge distribution: $\sup_{a \leq u \leq b} \frac{|W^0(u)|}{\sqrt{u(1-u)}}, 0 < a < b <1 $

If $W^0$ is a tied-down Wiener process (Brownian bridge) on the range $(0, 1)$, what is the distribution of \begin{equation} \sup_{a \leq u \leq b} \frac{|W^0(u)|}{\sqrt{u(1-u)}} \end{equation} ...
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1answer
127 views

Basic Martingale question

We know the following is a martingale, and is commonly used to represent Ito's Integral. If $W$ is a brownian motion. $ \int_0^T W(u) dW(u) = \frac{1}{2} W^2 (t) - \frac{1}{2}t, \ for \ \ T \ge t \ge ...