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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18 views

Holder continuity of Brownian Motion

Can any one help me in this question please: By using the law of iterated logarithm, I have to show that $\forall t \in [0, T]$ ; where T>0, $\exists A_{t} \in \mathcal{A}$ with $P(A_{t})=1$, a random ...
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21 views

To show that $\dim Z=1/2$, why do I have to show that $p\{\dim Z=1/2\}=1$?

Let $(B_t)_{t\geq 0}$ a standard Brownien motion. I have to show that $\dim Z=\frac{1}{2}$ where $Z=\{t\in [0,1]\mid B_t=0\}$. Why to do this, I have to show that $$\mathbb P\{\dim Z=1/2\}=1\ \ ?$$ I ...
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43 views

Brownian Motion hitting time is finite yet has infinite expectation?

I've read that a hitting time of a Brownian motion (defined as $T_a = \inf\{t\ge0:W_t=a\}$ where $W_t$ is a standard Brownian Motion, i.e. a Wiener process), has the following two properties, which I ...
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1answer
27 views

Reference request for stochastic processes on manifolds

I'm looking for some references on stochastic processes on manifolds. The more introductory the better. Thanks.
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38 views

Ito's formula application

Let $ \alpha, \beta \in R$ and define $$ N(t)=e^{\beta t} \cos(\alpha W (t)) $$ I need to use Ito formula to compute $dN(t)$ Suppose $\alpha$ is fixed. What should $\beta$ be so that $N$ is a ...
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35 views

Yet another application of Ito's formula

Question : Let $dW^4(t) $ be the sum of an ordinary integral with respect to time and an Ito integral. Where $W^4(t)$ are standard Brownian motion. I am trying to apply Ito's formula to this, say ...
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18 views

bromnian motion and use of Lebesgue's differentiation theorem

Let $M$ be a Brownian motion with $M_0=0$ and $V\in L(M)$. Use Lebesgue's differentiation theorem to prove that there exists a predictable process $H\in L(M)$ such that $V\cdot M$ and $H\cdot M$ are ...
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32 views

show that a function with brownian motion is a martingale

Let $B=(B^1,B^2)$ be a two-dimensional Brownian motion w.r.t. the Filtration $\mathcal{F}^B$. Show that $(M_t^2)_{t\in \mathbb{R}_{+}}:=(e^{B_t^1} \cos(B_t^2))_{t\in\mathbb{R}_{+}}$ I've tried it ...
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1answer
37 views

Integral representation $B_T^3$

I have to find a $F_t$ such that $B_T^3=E[B_T^3]+\int_0^T F_t dB_t$. I have shown by ito formula that $B_T^3=\int_0^T 3 B_s^2 dB_s+\int_0^T 3 B_s ds$. Could you please help me?
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21 views

Brownian martingale as time-space changed brownian

Let $M$ be a true real martingale adapted to some brownian motion $B$. What are the most generic conditions on $M$ to find a deterministic map $\Phi:\mathbb{R}_+\times\mathbb{R} \to \mathbb{R}_+\...
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2answers
39 views

Checking if $X(t) = \exp(t/2)\cos(W(t))$, with $W(t)$ a Wiener process, is a martingale

This is what I've done: Let $s < t$ and $F_t$ be a filtration adapted to $W(t)$ $$E[e^{t/2}\cos(W(t))|F_s] = e^{t/2} E[\cos(W(t)) - \cos(W(s)) + \cos(W(s))|F_s]$$ $$= e^{t/2} [E[\cos(W(t)) - \cos(...
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1answer
42 views

How to use the Markov property of Brownian motion

This is a problem from Durrett's probability with examples, exercise 8.2.1. It is not homework. The exercise states: Let $T_0 = \inf\{s > 0 : B_s = 0\}$ and let $R = \inf\{t > 1 : B_t = 0\}$. ...
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0answers
30 views

Conditional expectation of a hitting time of a Brownian motion and Laplace transform

I am trying to solve the following problem: Suppose B is a 1-dim Brownian motion, let $\mathcal{T}_a = inf\{t: B_t = a\}, \mathcal{T}_{a,b}=min\{\mathcal{T}_a,\mathcal{T}_b\}$. For $a < 0 < b$ ...
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88 views

Brownian motion on sphere proof?

proving the brownian motion on the sphere equation the stratonovich form differential equation $$\partial X=n(X)\times \partial B$$ the equation in ito's form becomes $$dX=n(X)\times dB+H(X)n(X)...
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1answer
32 views

Probability that a Wiener process is negative at 2 given that it was positive at 1

Let $W_t$ be a standard Wiener process, i.e., with $W_0=0$. If $W_1>0$, what is the probability that $W_2<0$? This is my attempt: we want to determine the conditional probability $$\mathbb P(...
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9 views

Upper bound involving simple Ito process

Let $(B(t),\{\mathcal{F}_t \})$ be one-dimensional Brownian motion. Suppose that $\sigma(t,ω)$ is a $\mathcal{F}_t$-adapted process satisfying $|\sigma(t,ω)| ≤ R$, for all $t$ and $w$. I was ...
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33 views

Laplacian in spherical coordinates - brownian motion

Consider the Laplacian equation on the unit sphere, for a vector $f$. $\theta$ is polar angle, and $\phi$ is azimuthal angle. The Laplacian in spherical coordinate is : $$ \Delta f = {1 \over r^2} {\...
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1answer
35 views

Brownian hitting time of a closed set

I am trying to prove that the first hitting time of a closed set H by a Brownian motion is a stopping time. I have found a proof that states: $$\{\mathcal{T}\leqslant t\} = \bigcap_{n=1}^{\infty}\...
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40 views

Simple application of Donsker's theorem

I am trying to do exercise 5.15 in Moerter's book "Brownian Motion". It seems quite easy, but I can't solve it somehow: Suppose $S(j)_j$ is a SRW on the integers, started at zero. Show that: $$ \frac{...
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30 views

BM hitting times with exponential killing process

Assume a BM in 3d domain (infinite) with a small absorbing subdomain (cube, sphere, ect), centered at point $p_s=(x_s,y_s,z_s)$. BM starts at point $p_0=(x_0,y_0,z_0)$ and when it riches the subdomain ...
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20 views

Coupling Brownian Motions

I want to simulate three freight rate indices which are naturally correlated. The freight rate dynamics ($X$) can be modeled as a geometric Brownian motion: $dX_{t} = \mu X_{t}dt + \sigma X_{t}dW_{t}$...
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1answer
19 views

Distribution of the first exit time of a one-dimensional diffusiom/ Brownian motion

I have a one-dimensional diffusion on $[0,1]$ and I need to calculate the distribution of the first exit time of the interval $(-\epsilon,\epsilon)$ for an $\epsilon > 0$. A good first step would ...
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17 views

How to find the mean and variance of a stochastic integral?

If $B(t)$ is a standard Brownian motion, let $Z(t)= \int_{0}^{t} s^2 dB(s)$. I want to find the mean and variance of Z(t). It is given that $Z(t)$ is Gaussian process. My approach for finding the ...
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0answers
22 views

Intuition behind “stochastic orthogonality”

Whilst doing an exercise on the Brownian Motion on a sphere I came across this identity: $$ \langle Z\times B,Z\times B\rangle = 2|Z|^2dt $$ where $\times$ denotes the cross product and $Z$ is a ...
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1answer
34 views

Prove that $\text{lim}_{\Delta t} \rightarrow 0$ of the transition PDF of a std Weiner process is 0

The transition probability density function of the standard Wiener process is: $$ f(x_2,t_2|x_1,t_1) = \frac{1}{\sqrt{2 \pi (t_2-t_1)}}e^{-\frac{(x_2-x_1)^2}{2(t_2-t_1)^2}} $$ I know that if Markov ...
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1answer
29 views

Show that $p(t_0\ \text{is a local maximum for}\ B)=0$.

Let $B$ a Brownian motion. Show that for all $t_0$,$$p\{t_0\text{ is a local maximum for }B\}=0$$ but a.s. local maximal are a countable dense set in $(0,\infty )$. For the first part, $$p\{t_0\text{...
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1answer
16 views

Einstein's number of particles that experienced a certain shift explanation

I am reading a Gardiner's Stochastic Methods handbook and I am wondering about the meaning of the following (this is the very beginning of the chapter): $dn = n \phi(\Delta) d \Delta$ This is ...
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0answers
32 views

Proof of normal distribution property used in Levy's construction of the brownian motion?

I have been trying to follow the construction of Brownian motion by Levy. I need a property about the conditional distribution of the Brownian process in an interior point of an interval given its ...
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31 views

Can we apply an Itō formula to the solution of a SPDE?

Let $V\subset H\subset V^\ast$ be a Gelfand triple $(\Omega,\mathcal A,\operatorname P)$ be a probability space and $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $(W_t)_{t\ge 0}$ be a ...
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26 views

Long term behavior of Brownian Motion

Let $(B_t)_{t \geq 0}$ be a Brownian motion. The objective is to prove that \begin{align*} \limsup_{t \to \infty} \frac{B_t}{\sqrt{t}} = \infty. \end{align*} By the scaling property of Brownian ...
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1answer
33 views

Why does the Borel-Cantelli lemma finish the job? - Law of Large Numbers Brownian Motion

The objective is to prove that \begin{align*} \text{$\lim_{t \to \infty} \frac{B_t}{t} =0 \qquad$ a.s.} \end{align*} By the strong Law of Large Numbers, we have that: \begin{align*} \text{$\lim_{t \...
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17 views

Brownian motion: Why $\xi_n$ and $\xi_m$ are independent.

Let $\{B_t\}$ a Brownian motion, $n\neq m$ and $$\xi_n=\sup_{s\in [n,n+1]}|B_s-B_{\lfloor s\rfloor}|\quad \text{and}\quad \xi_m=\sup_{t\in [m,m+1]}|B_t-B_{\lfloor t\rfloor}|.$$ We suppose WLOG that $...
3
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1answer
107 views

Discrete and continuous Girsanov

I'm trying to write a proof of the Girsanov theorem based on a discrete version of it. Discrete version Suppose that I have a random vector $X$ and two equivalent probability measures $\mathbb{P}, \...
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0answers
18 views

If $B_t - B_s, \ 0\leq s < t,$ is normally distributed, there are constants $C_n, \ E|B_t - B_s|^{2n}=C_n|t-s|^n$

I am working on the following problem: Show that if $B_t - B_s, 0 \leq s < t,$ is normally distributed with mean zero and variance $t-s$, then for each positive integer $n$ there is a positive ...
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1answer
27 views

Quadratic Variation of Wiener's process

I know I'm wrong, but I still don't understand why can't this operation be performed: $$ \sum_{j=0}^{n-1}(W(t_{j+1})-W(t_j))^2\le \max[W(t_{j+1})-W(t_j)]*(W(T)-W(0) )$$ which would have a limit of $...
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0answers
25 views

Show that $W(t)$ is almost surely non-differentiable at $t=0$

Show that $W_t$ is almost surely non-differentiable at $t=0$. Of course, $W(t)$ denotes a standard Wiener process. It is enough to show that $$P(\{\omega : \exists \epsilon>0 \: \forall \delta &...
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0answers
24 views

Compare moments of $\int_0^t h(B_s) ds$ and $\int_0^t h(\sqrt{s}Z)ds$ for $(B_t)$ Brownian motion and $Z$ standard normal

If we let $B_t$ be a standard Brownian motion and $\sqrt{t}Z$, where $Z$ is our standard normal random variable, we know that they have the same distribution. However, how can I show that the process ...
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38 views

How to find the variance of $\int_0^t B_s^2 ds$ where $B_s$ is a standard Brownian motion random variable?

I am trying to find the variance of $\int_0^t B_s^2 ds$ where $B_s$ is a standard Brownian motion random variable. My approach is to represent the integral as a sum. However, I am not sure how this ...
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25 views

Proving if $W(t)$ is a weiner process, then $W^2(t)$ is also a Weiner process [duplicate]

I'm trying to solve this question: For a stochastic process to be a Weiner Process it must have these properties: $W(0) = 0$ so $W^2(0) = 0$ $E(W(t)) = 0$ but $E(W^2(t)) = t$ I think this is enough ...
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0answers
19 views

What is meant by local time of BM on the boundary $\partial D$?

I'm familiar with local time $L_t^a$ at level $a$ for a 1-D Brownian motion $B$. I'm reading this paper which talks about a 2D Brownian motion $B$ in a bounded domain $D$ that gets reflected when it ...
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0answers
27 views

Probability of hitting a barrier

We have a stochastic process $ Y_t= \alpha t+ W_t$ where W is a standard brownian motion. Is there a way to calculate the conditional probability with respect to $Y_1$ for this process to hit a ...
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1answer
47 views

Hitting times for Brownian Motion (2)

In this post there is shown that for a standard Brownian motion $\mathbb{E}[\tau^p]<\infty$ for all $p \geq 1$, where \begin{align} \tau = \inf\{t \geq 0 : B_t = \alpha \ \ \text{or}\ \ B_t=-\beta\...
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1answer
50 views

How can we identify $\omega\in\Omega$ with a path of Brownian motion $t\rightarrow B_t(\omega)$?

In the Stochastic Differential Euqations written by Oksendal(see page 12), As we shall soon see, the paths of a Brownian motion are (or, more correctly, can be chosen to be) continuous, a.s. ...
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25 views

PDE for Brownian Bridge Expectation?

Let $\displaystyle Y(t)=\int_0^t v(s)ds+B(t)$, where $B(t)$ is the standard Brownian motion and $v(t)$ a deterministic function. Compute $m(t,y):=\mathbf E\Big[\max\limits_{s\in[0,t]} Y(s)\big|Y(t)=y\...
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2answers
51 views

Hitting times for Brownian motions

Let $B$ be a standard Brownian motion and let $\alpha, \beta > 0$. Let \begin{align} \tau = \inf\{t \geq 0 : B_t = \alpha \ \ \text{or}\ \ B_t=-\beta\}. \end{align} It can be shown by defining ...
3
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1answer
51 views

$X_t=\int_{0}^{t}(a_{0}+a_{1}\frac{u}{t}+\ldots+a_{n}\frac{u^{n}}{t^{n}})dB(u)$ is a Brownian motion for suitable non-zero constants $a_0,\ldots,a_n$

Let $B(t)$ be brownian motion. Show that for any integer $n \geq 1$, there exist nonzero constants $a_{0},\ldots,a_{n}$ such that $X_{t}=\int_{0}^{t}(a_{0}+a_{1}\frac{u}{t}+\ldots+a_{n}\frac{u^{n}}{t^{...
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0answers
41 views

Last exit time of Brownian motion

I am trying to show that the last exit time of Brownian motion is a random variable, i.e. for $\tau$ defined as $$\tau = \sup\{t > 0 : W_t = 0\}$$ it holds that $\{\tau < t\} \in \mathcal{F}$ ...
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0answers
55 views

What is the solution to this graduate-level statistics problem?

I'm baffled as to how to explicitly solve this problem... I would normally just plug in problem-specific values and use Monte Carlo simulation to solve something complicated like this, but my ...
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1answer
38 views

Brownian Motion with rescaled time as an Ito process

I have a seemingly simple question that has me stumped. Suppose $(B_t)_{t\geq0}$ is a Brownian motion, and consider its rescaled version $(B_{\alpha t})_{t\geq0}$ for some $\alpha>0$. It seems ...
2
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0answers
30 views

Laplace transform of survival probability for stochastic diffusion

Let $Y_t$ be a killed process defined by \begin{eqnarray} Y_t = X_t \quad \mbox{if } t<\xi,\\ Y_t = 0 \quad \mbox{if } t\geq\xi. \end{eqnarray} where $\xi$ is a random time such that $$ \xi=\inf \...