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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General question about expected value of brownian motion

I was wondering if anyone could tell me a little about expectation of brownian motions and how it is connected with normal distribution. I know that B(t) is N(mean*t,variance*t) and that the expected ...
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54 views

A Special case of geometric Brownian motion

I'm sitting with a special case of GBM where $B(t)$ is Brownian motion with drift $\mu$ and variance $\sigma^2$. I want to find the expected value of $A(t)$, where $A(t)$ is given by ...
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79 views

Independence in Brownian Motion

I've read two times in different lecture notes that for a Brownian Motion $(B_s)_{s\ge 0}$ and $t<u$ the random variable $B^t_{u}:=B_{t+u}-B_t$ is independent from ${\cal F}_t:=\sigma\{B_s:s\le ...
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50 views

$B(t)$ is brownian motion. I want Find $d(M(t))^2$,where $M(t)=e^{B(t)-\frac{t}{2}}$,

let $B(t)$ is brownian motion. Find $d(M(t))^2$,where $M(t)=e^{B(t)-\frac{t}{2}}$,
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46 views

I want Find $d(\frac{X(t)}{Y(t)})$ where $B(t)$ is a brownian motion and $X(t)=tB (t)$ and $Y(t)=e^{B(t)}$.

Let $B(t)$ is a brownian motion and $X(t)=tB (t)$ and $Y(t)=e^{B(t)}$. Find $d(\frac{X(t)}{Y(t)})$ Thanks for help.
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81 views

Fake Brownian Motion

Does there exist a martingale which has Marginal distributions same as Brownian Motion marginals but the process itself not being Brownian motion? Any references are highly appreciated. Thanks.
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170 views

Time scaling of Brownian motion

Let $(B_t)_{t\geq 0}$ be a standard Brownian motion and $A_t$ be an increasing continuous process adapted to the filtration generated by the Brownian Motion and $A_0 = 0$. I am trying to prove ...
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1answer
53 views

Wiener process analytic expression from geometric brownian motion

The solution to the SDE $dx= -kx\ dt + cx \ dW$ is $x(t) = x_0 e^{(c - k^2/2)t}e^{-k W}$ with mean $\langle x(t) \rangle = x_0 e^{(c - k^2/2)t}$ where $W(t)$ is the Wiener process. Im ...
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1answer
92 views

The limit supremum of a function involving Brownian motion

I would like, for some $\delta>0$ and a Brownian motion $B$, to calculate $\displaystyle\limsup_{t\to\infty}\left(\exp\left( (1+\delta)t\right)\cdot\exp\left(-B_t-\frac{t}{2}\right)\right)$ ...
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144 views

Must $n$ independent Wiener processes be simultaneously positive at some time?

Consider $n$ independent one-dimensional Wiener processes $(W_i)_{1\leqslant i\leqslant n}$. Is there with probability $1$ some time $t\in[0,1]$ such that $W_i(t)>0$ for every $1\leqslant ...
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1answer
216 views

Exercise 3.3.25 of Karatzas and Shreve

This is the Exercise 3.25 of Karatzas and Shreve on page 163 Whith $W=\{W_t, \mathcal F_t; 0\leq t<\infty\}$ a standard, one-dimensional Brownian motion and $X$ a measurable, adapted process ...
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48 views

When is a continuous path stochastic process be representable as diffusion or Ito process?

When can a continuous path (Markovian) stochastic process in one dimension be represented as an Ito or a diffusion process? What are the examples when it can not be?
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1answer
63 views

I want to show $E(B(t)-B(s))^4=3(t-s)^2$

Let $B(t)$ and $B(s)$ are brownian-motion I want to show $$E(B(t)-B(s))^4=3(t-s)^2$$ thanks for help.
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1answer
90 views

Problem 3.2.28 of Karatzas and Shreve

It's the Problem 2.28 of Karatzas and Shreve on Page 147: Let $M=W$ be standard Brownian motion and $X\in\mathcal{p}$. We define for $0\leq s<t<\infty$ $$\zeta_t^s(X)\triangleq\int_s^t X_u ...
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1answer
57 views

Distribution of Difference of Independent Random Variables

Usually in the development of the theory of Brownian motion, one makes the assumption that $X_t$ (the coordinate functions on $(\mathbb{R}^*)^{[0,\infty)}$). have normal distributions with mean $0$ ...
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1answer
103 views

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 ...
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1answer
60 views

Distribuation Max - Min of Brownian motion

I'm looking for the distribuation of $M_X(t) - m_X(t)$ of the brownian motion and not the joint distribuation. where $m_X(t) = \min\limits_{0\leq s\leq t}X(s)$ and $M_X(t) = \max\limits_{0\leq s\leq ...
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2answers
94 views

Ito's Isometry for three factors

Ito's Isometry states the following: If $\{W_t\}_{t\ge0}$ is a Brownian motion and $\{\phi_t\}_{t\ge0},\{\psi_t\}_{t\ge0}$ are two non-anticipative piecewise-continous processes with $\mathbb ...
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0answers
109 views

Verifying a standard Brownian Motion?

Let $\{X_t, t\ge 0\}$ be a standard Brownian motion process. For a fixed positive number s and all $t\ge 0$, we define $Y_t = X_{t+s} - X_s$. Is $\{Y_t, t\ge0\}$ a standard Brownian motion? Attempt: ...
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416 views

$\int_0^tB_s^2\ dB_s$ - Gaussian Process and independent increments?

For $(B_t)_{t\ge0}$ a standard Brownian motion (Wiener process) define the stochastic process $X_t:=\int_0^tB_s^2\ dB_s$. I am currently trying to assess if $(X_t)_{t\ge0}$ is a Gaussian process and ...
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1answer
175 views

Expectation of product of correlated Brownian motions at different time points

Given the information about the correlation of two Brownian motions as $E[dW_1 dW_2] = \rho dt$ and knowing that $E[W_1(t)W_1(t')] = \min(t,t')$, I want to compute $E[W_1(t)W_2(t')]$ I interpret ...
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1answer
256 views

Blumenthal 0-1 law

Let $(B_t)$ be a Brownian motion. Consider the event : $B(n)>a \sqrt n $occuring infinitely often. I want to prove that this event has probability 1. we can see that, by rescaling property, ...
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215 views

Blumenthal zero-one law

How to prove $\limsup\limits_{n \to \infty} \frac{1}{\sqrt n}B(n) = +\infty$ using Blumenthal zero-one law, where B is a Brownian motion?
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1answer
128 views

differentiability of brownian motion

for a fixed $t \in [0, \infty)$, I have to show that $ \mathbb{P} (D^+W_t = + \infty$ and $D_+W_t = -\infty )$, where $D^+$ (and $D_+$) denotes the upper right-hand derivative (and respectively the ...
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1answer
104 views

Time-invariance and spatial-invariance of a stochastic process

Many stochastic processes have independent and stationary increments, i.e. let $(X_t)_{t\ge 0}$ be a stochastic process on a probability space $(\Omega,\mathcal{F},\mathbb{P})$, then $X_t - X_s ...
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2answers
83 views

change of sign for brownian motion

for a fixed $\epsilon$ $> 0$, I want to show that almost surely (ie with probability 1), a standard brownian motion $W_t$ would change sign over [o,$\epsilon$ ]. I thought about defining a random ...
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1answer
90 views

Law of Large numbers using Brownian limit

Given a standard Brownian motion $\{B_t;0 \leq t < \infty \}$, we know that $\lim_{t \to \infty}\frac{B_t}{t} = 0$ a.s. I am interested to know if we can prove Strong Law of Large Numbers for any ...
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181 views

What is the probability a random walk hits x before it hits y?

This problem was motivated by my bitcoin trading and recalling some of my math education back in the day. I thought I'd ask people who know this much better than I... Suppose there is a continuous, ...
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126 views

Law of large numbers for Brownian Motion

Let $\{B_t: 0 \leq t < \infty\}$ be standard Brownian motion and let $T_n$ be an increasing sequence of finite stopping times converging to infinity a.s. Does the following property hold? ...
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1answer
124 views

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
140 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
44 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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1answer
163 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
85 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
87 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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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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72 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
194 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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29 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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51 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
375 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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1answer
378 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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166 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
200 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
149 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
683 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
75 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
143 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
122 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 ...