1
vote
0answers
45 views

Donsker for randomly stopped processes

A question regarding Donsker's invariance principle. Donsker states that if $X_1, X_2, ...$ are independent and identically distributed with mean $0$ and variance $\sigma^2$ and if $S_t^n$ is the ...
2
votes
1answer
51 views

Is the graph of a Brownian motion over an interval measurable?

Let $n \in \mathbb{N}_1 := \{1, 2, \dots\}$ and let $B:\Omega \times [0, \infty) \rightarrow \mathbb{R}^n$ be a standard, $n$-dimensional Brownian motion over the probability space $(\Omega, ...
1
vote
1answer
42 views

Convergence to integral: $\sum_{k=1}^{k_n}f\left(B_{t_{k-1}^{(n)}}\right)\left(B_{t_{k-1}^{(n)}}-B_{t_{k}^{(n)}}\right)^2 \to_p \int_0^Tf(B_t)dt$

The problem goes: Let $(B_t)$ be a standard Brownian motion, and $f:\mathbb{R}\to\mathbb{R}$ be continuous. Show that if $T>0$ and $(P_n)$ is a sequence of partitions of $[0,T]$: ...
3
votes
1answer
62 views

Integrating brownian motion times exponential function

I am trying to calculate $$\int_0^tB_se^{\lambda s}ds$$ but I am unsure of how to start the computation. The motivation behind this is that I read (and am now trying to prove) that ...
3
votes
0answers
99 views

An exercise from Revuz, Yor; equality in distribution of 2 integrals.

Here is the exercise I have been struggling to solve. It is taken from this book by Revuz and Yor: link. Here is the full text of the problem ( Exercise 3.32, chapter 4). Exercise (3.32). Let $B$ and ...
0
votes
0answers
18 views

Reflected borel sets are worse traps for Brownian paths.

This is for a research project. I am trying to prove that given a borel set, it's reflected version will have a lower Wiener measure of brownian paths intersecting it. In this paper they elaborate ...
1
vote
1answer
62 views

An application of the strong Markov property in the proof of the connection between Brownian motion and harmonic functions

Let $U$ be an open, connected set in $\mathbb{R}^n$ and let $(B(t))_{t \geq 0}$ be an $n$-dimensional Brownian motion with start at $x \in U$ and let $\overline{B_x(\delta)}$ be the closed ball about ...
7
votes
2answers
169 views

A planar Brownian motion has area zero

I'm looking for proofs of Paul Lévy's theorem that a planar Brownian motion has Lebesgue measure $0$. I know of only two proofs: one is in Lévy's original paper (Théorème 12, p. 532) and the other is ...
1
vote
2answers
59 views

Does a Brownian motion remain in any given open set for a given interval of time with positive probability?

Let $B$ be a standard $d$-dimensional Brownian motion. Given $b>a>0$ and an open ball $U$ in $\mathbb{R}^d$, I want to be able to comment on the probability that $B$ remains in $U$ during the ...
3
votes
1answer
115 views

Proving the reflection principle of Brownian motion

The reflection principle of Brownian motion states that Brownian motion reflected at some stopping time $\tau$ is still a Brownian motion. The proof found in Mörters & Peres (as well as in ...
1
vote
0answers
72 views

Tail of hitting times for Brownian motion on the circle

For $y\in \mathbb R/\mathbb Z$ and $\varphi\in C([0,\infty);\mathbb R/\mathbb Z)$ let $T_{y}(\varphi) \ := \ \inf\{t>0: \varphi_t = y \} \ \ \ $ (first time the path $\varphi$ hits $y$) ...
0
votes
1answer
35 views

Invariance of Brownian motion under orthogonal transformations

Let $\left(B_t\right)_{t \in [0,\infty)}$ be an $n$-dimensional Brownian motion with start at $x \in \mathbb{R}^n$, and let $A$ be an orthogonal $n \times n$ real matrix. I'm trying to show that $AB$ ...
1
vote
0answers
14 views

Hitting time for a planar diffusion

Let $A$ be an open subset of $\Bbb R^2$, and let us consider a diffusion $\mathrm dX_t = f(X_t)\mathrm dt + g(X_t)\mathrm dW_t$ where $f$ and $g$ are globally Lipschitz continuous maps. Suppose I am ...
2
votes
1answer
42 views

The probability of a Brownian motion's tail event is unaffected by the starting point

Consider the measurable space $\left(\mathbf{C}\left[0,\infty\right), \mathcal{B}\left(\mathbf{C}\left[0,\infty\right)\right)\right)$ and the stochastic process $\left(X_t\right)_{t \in ...
2
votes
1answer
53 views

Diffusion processes

I am trying to work out a problem to which I have not found similar solutions on the website. Perhaps you can help me out. Let $X = (X_t)_{t\geq0}$ be a non-negative diffusion process which solves ...
2
votes
0answers
29 views

Local time of fractional Brownian motion

For BM, there is a downcrossing representation of the local time at 0. Namely, $L_t(0)=\lim_2 (b_i-a_i)D(a_i,b_i,t)$, where $D$ is the number of downcrossing between level $b_i$ and $a_i$. I am ...
1
vote
2answers
50 views

$4^{Brownian(t)}$ martingale proof

Let $B(t)$ a Brownian motion. I like to prove that $4^{B(t)}$ = martingale I rewrote the expression into an exponential form (like $\exp(\ln(4) B)$), but then I don't know how to proceed.
5
votes
2answers
80 views

Estimating the maximum of a Brownian motion over the unit interval

Let $\left(B_t\right)_{t \in \left[0,\infty\right)}$ be a standard Brownian motion over the probability space $\left(\Omega, \mathcal{A}, P\right)$. For each $x \in \left(0, \infty\right)$, give an ...
1
vote
1answer
42 views

First hitting time in the one-dimensional case by solving a boundary value problem

If have a question about section 3.1 in the paper Kramers' law: Validity, derivations and generalisations by Nils Berglund. (See http://arxiv.org/abs/1106.5799 page 7 - 9) On page 8 it says, that ...
6
votes
2answers
118 views

What is the importance of the infinitesimal generator of Brownian motion?

I have read that the infinitesimal generator of Brownian motion is $\frac{1}{2}\small\triangle$. Unfortunately, I have no background in semigroup theory, and the expositions of semigroup theory I have ...
2
votes
2answers
66 views

A Brownian motion $B$ that is discontinuous at an independent, uniformly distributed random variable $U(0,1)$

Suppose that $\left\{B\left(t\right): t \geq 0\right\}$ is a Brownian motion and $U$ is an independent random variable, which is uniformly distributed on $\left[0,1\right]$. Then the process ...
2
votes
1answer
23 views

Strong approximation of a brownian motion path by a polygonal path

Consider an SBM $(B_t)_{t\geq 0}$. Now we can obtain a polygonal path on $[0,n]$ by joining the integral points $B_0, B_1, \ldots, B_n$ with segments and call this path $B^{n}_t$. Now I want to bound ...
0
votes
1answer
36 views

Proof that finite-dimensional Wiener process distributions are Gaussian

I have to prove that finite-dimensional Wiener process distributions are Gaussian and calculate them. How should I start? I know the definition and properties of Wiener process.
0
votes
0answers
51 views

brownian motion and stopping time

I have an exercise about Brownian motion which I don't understand completely. Let $(B_s)_{s\geq0}$ be a standard real Brownian motion. For $t > 0$, we define the random times $g_t ...
1
vote
0answers
48 views

An application of the Dambis-Dubins-Schwarz theorem. Is my argument correct?

I attended a lecture today, in which the professor went through an example with a lot of tedious calculations to show something which I'd think would follow directly from the Dambis-Dubins-Schwarz ...
4
votes
1answer
60 views

Laws and Moments of two dimensional brownian motions

I am a bit rusty on this. So let us consider the following two dimensional standard Brownian motion issued from zero defined on the probability space $(\Omega, \mathcal{F},\mathbb{P})$ (note that, in ...
1
vote
1answer
73 views

Quadratic covariation of Itô processes

I haven't found any similar question in the forum, so I trust some of you will find this thought-provoking (at the very least). Perhaps you can help me. Let's consider first the two following ...
2
votes
1answer
73 views

Show that this is a stopping time

Show that $\sigma=\inf \{ t\ge 0 : |B_t|= \log t \}$ is a stopping time with respect to $(\mathcal F_t^B)_{t\ge0}$. I've been trying to put the set $\{\sigma\le t\}$ equal to a countable union and ...
0
votes
1answer
48 views

Show that a process is no semimartingale

_Hello everyone! I got a little question about how to show that the process $X_t:=|B_t|^{\frac{1}{3}}$ is NOT a semimartingale. So far I tried to apply Ito. Since if $X_t$ was a semimartingale so is ...
3
votes
1answer
179 views

Expectation of stochastic integrals related to Brownian Motion

I'm trying to solve a problem that's now doing my head in a bit. I'll share with you the question and let's see if somebody can shed some light into the matter: Let B be a standard Brownian Motion ...
0
votes
0answers
57 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 ...
1
vote
1answer
50 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.
4
votes
0answers
90 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 ...
3
votes
1answer
44 views

The limit supremum of a function involving Brownian motion

I would like, for some $\delta>0$ and a Brownian motion $B$, to calculate $\displaystyle\limsup_{t\to\infty}\left(\exp\left( (1+\delta)t\right)\cdot\exp\left(-B_t-\frac{t}{2}\right)\right)$ ...
1
vote
1answer
86 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 ...
2
votes
1answer
57 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.
1
vote
1answer
52 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 ...
2
votes
1answer
37 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 $0$ ...
2
votes
1answer
64 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 ...
1
vote
1answer
50 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? ...
1
vote
1answer
72 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 ...
1
vote
1answer
128 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 ...
0
votes
2answers
69 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 $\int_{0}^{a}B(t)dt\sim N(0,\frac{a^3}{3})$
1
vote
2answers
47 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 ...
1
vote
1answer
53 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 ...
3
votes
0answers
42 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 $x_t$ ...
1
vote
0answers
37 views

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 ...
0
votes
1answer
44 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, ...
1
vote
1answer
90 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 ...
1
vote
2answers
104 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 ...