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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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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36 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
37 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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21 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
21 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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20 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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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
35 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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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
35 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 $...
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1answer
124 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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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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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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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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21 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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29 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
52 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
52 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
58 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 ...
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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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44 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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61 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
39 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 ...
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31 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 \...
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0answers
27 views

Expected time when Brownian motion leaves an interval

Let $S_t$ be standard Brownian motion (or a Wiener process) in one dimension. How do I formally derive the expected time that $S_t$ will leave a given interval $[-x, y]$ for some $x, y > 0$, given ...
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1answer
36 views

Deriving heat equation from brownian motion

Today my prof gave me an equation of random walk: $$p(x_i,t+\Delta t)=\frac{1}{2}(p(x_i-\Delta t)+p(x_i+\Delta t))-p(x_i,t)$$ Using this he get$$P_t=P_{xx}$$ when $\Delta t<<1$ But how and what'...
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1answer
44 views

Martingale property for two stochastic processes

Let $(\Omega,F,P)$ be a probability space with filtration $\left\{F_{t}\right\}_{t\geq 0}$ generated by one dimensional Brownian motion $(B_{t})_{t\geq 0}$ defined on $(\Omega,F,P)$, assume that $\...
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1answer
23 views

Applying ito integral to a specific example

Using the ito integral: $$x_t=\int_0^t b(s,x(s)) \, ds+\int_0^t \sigma (s,x(s)) \, dB(s)+x_0$$ I want to solve this: I have $$g(s,n(s))=\log(n(s))$$ Using ito's lemma taking the derivative I get ...
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1answer
17 views

GBM for the stock prices

Given the following SDE: $$ dS = \mu S dt + \sigma S dW $$ Why when we write this in discrete form: $$\Delta S = S_{i+1} - S_i = \mu S_i \Delta t + \sigma S_i \phi \sqrt{\Delta t}$$ The indices on ...
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1answer
85 views

Explaining a simple observation on Terry Tao's blog about the Wiener process

Quoting Terry Tao's blog: A simple but fundamental observation is that $n$-dimensional Brownian motion is rotation-invariant: more precisely, if $(X_t)_{t \in [0,+\infty)}$ is an $n$-...
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14 views

Geometrical Brownian motion Passage time

Recently I have been self-studying stochastic analysis. One of the exercises was to calculate the probabilty of Brownian motion reaching certain level before time T given that W(t)=x. This wasn't that ...
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1answer
53 views

Solve the stochastic differential equation $dX_t = \alpha X_t \, dW_t + \sigma X_t \, dt$

We want to solve: $dX_t = \alpha X_t dW_t + \sigma X_t dt$ where the initial condition $X_0$ is given and $\alpha$, $\sigma$ are constant. The solution goes as follows - We rewrite the ...
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1answer
65 views

Measurability of the zero-crossing time of Brownian motion

I have the following random time $\tau = \inf\{t > 0: W_t = 0\}$ where $(W_t)_{t\geq 0}$ is Brownian motion with almost surely continuous paths and $W_0 = 0$ a.s. I need to prove that $\tau$ is ...
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68 views

Proving how to reduce a Brownian walk on a plane to a line (2D to 1D)

I have a Brownian motion on a plane and would like to find the time of when it is expected to hit a set of parallel lines, i.e the hitting time. In order to do so, I understand that I can reduce the ...
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35 views

Brownian motion - independence

I have not so difficult task - For Brownian motion $W(s)-W(t)$ is independant of $\sigma$-algebra $F(t)$ $0\leq t<s$. My goal is to show that for $0\leq t<s<u$, $W(u)-W(s)$ is also ...
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1answer
20 views

What is a general Brownian Motion?

This might be a dumb question, but no textbook ever defines what a "Brownian Motion" is, just what a "Standard Brownian Motion." I always assumed that a Brownian Motion is any random variable that can ...
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29 views

Brownian motion checking

$W(t)$ is a Brownian motion and $c>0$. I need to verify that 1)$X(t)=W(c+t)-W(c)$ and $X(t)=cW(t/c^2)$ are Brownian motions. 2) $Z(t)=tW(1/t)$ can be showm as $lim_{t-\rightarrow\infty} Z(t)=0$....
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2answers
44 views

If $B_t$ is a standard brownian motion process, is $B_t^2 - \frac{t}{2}$ a martingale w.r.t. brownian motion?

If I have that $B_t$ is a standard brownian motion process, is $B_t^2 - \frac{t}{2}$ a martingale w.r.t. brownian motion? I know that $B_t^2 - t$ is but can't see it for the latter.
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14 views

Showing time inversion of a Brownian Motion $X_t = tB_{1/t}$ is continuous at $t=0$ USING the fact $X_t$ is BM on $\mathbb{Q}$? [duplicate]

I am reading the following paper on a rigorous construction of Brownian Motion: Brownian Motion. In the paper, they give a peculiar proof of the fact that the time inverted Brownian Motion is ...