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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2
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2answers
225 views

The 1-dimensional Hausdorff measure of a curve in the plane

For a set $X\subseteq\mathbb{R}^2$, let $H^1(X)$ be its 1-dimensional Hausdorff measure. Suppose $X$ is a regular curve (say, a graph of a continuous function $f:\mathbb{R}\to\mathbb{R}$). In that ...
0
votes
0answers
13 views

Maximum of Brownian Motion and a constant

I am interested in the distribution of $Z(t) = \max\{B(t),m\}$ where $B(t)$ is a standard Brownian motion and $m$ is a constant. By distribution, I mean the distribution of $Z(t)$ for a given $t$. I ...
1
vote
1answer
11 views

Conditional expectation: when does $X_t=E[X_t\mid \mathcal{F}_s]$ for $s<t$

I came across a calculation (1$^\circ$ calculation, 2$^{nd}$ step) that stated, for $s<t$ $$E[B_s(B_t^2-t)]=E[B_sE[(B_t^2-t)\mid\mathcal{F}_s]]$$ I know the expectation here is zero, however, I ...
0
votes
0answers
9 views

Invariance of harmonic function

Let $D^c\in\mathbb{R}^d$ be a compact set with positive Lebesgue measure, $W_t$ is a Brownian motion on $\mathbb{R}^d$ and $u(x)$ is a harmonic function on $D$. Define $$\tau_D=inf\{t>0|W_t\notin ...
5
votes
0answers
75 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 ...
1
vote
0answers
16 views

Brownian motion conditional expectation: $E\left[(B_s-B_t)^3\mid\mathcal{F}_t\right]$

Compute: $E\left[(B_s-B_t)^3\mid\mathcal{F}_t\right]$, $s>t$. $B_t$ is standard 1D Brownian motion, and $\mathcal{F}_t=\sigma(B_t)$. Here is my attempt: $$ \begin{aligned} ...
0
votes
1answer
39 views

How would you simulate Brownian motion with a die?

You can simulate Brownian motion on $[0, 1]$ for instance by splitting it into $K$ intervals and at each time $k \Delta t$ add $N(0, \Delta t)$ to your running total, where $\Delta t = 1/K$. If you ...
2
votes
0answers
30 views

Hitting Times for Brownian Motion - Levy Process?

Let $X$ be a Brownian motion and let $$H_a = \inf\{ s \ge 0 \mid X_s = a \} \;\ \text{and} \;\ S_a = \inf\{ s \ge 0 \mid X_s > a \}.$$ Now, I've shown that $H_a$ and $S_a$ are equal almost surely ...
3
votes
1answer
672 views

Hermite Polynomials and Brownian motion

I am asked to prove the following : Let $B_t$ be a standard brownian motion. The $n$th Hermite polynomial is $\displaystyle H_n(t,x)=\frac{(−t)^n}{n!} e^{x^2/(2t)} \frac{d^n}{dx^n}e^{-x^2/(2t)}$. ...
1
vote
0answers
33 views

Some Kind of Generalized Brownian Motion

Let $\displaystyle X(t) = \int_0^t f(s)dB(s)$ where $B(t)$ is a Brownian motion and $f(t)\in L^2[0,1]$. What is a simple representation for $Y(t):=(X(t)|X(1))$ in terms of $B(t)$? Note, I am not ...
1
vote
1answer
31 views

Maximum Likelihood Estimation of Brownian Motion Drift

I'm looking at times series of stock movements over 10 minute windows, and am trying to measure the "trend" of these movements. Method A is to simply calculate $\Delta P$, the difference between the ...
0
votes
1answer
20 views

Martingale and local martingales

I have to show that $e^{B_t^1}cos(B_t^2)$ is a martingale ($B=(B^1,B^2)$ is a two-diemensional Brownian Motion). I used Ito's formula and got $e^{B_t^1}cos(B_t^2)=1+\int_0^t ...
1
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0answers
12 views

What is skew-product decomposition?

What is skew-product decomposition of Brownian motions referring to this paper Pauwels ,Rogers
3
votes
0answers
17 views

Expectation related to Wiener process using strong Markov property

Can you help me to understand a result I found in Krylov's book "Introduction to stochastic calculus". First, I will introduce some notations: $w_t,t\ge 0$ denotes a Wiener process. ...
2
votes
0answers
21 views

Expectation of an Exponentiated Integral of a Brownian Bridge

Given a Brownian bridge $X(t)$ where $X(0)=0$ and $X(1)$ equal to some given constant. What is $\displaystyle \mathbf E\Big[\exp\Big(\int_0^1X(t)dt\Big)\Big]$? I suppose I can always discretize the ...
-1
votes
1answer
17 views

Correlation between a brownian motion and its increment [on hold]

Let $W(t)$ be a brownian motion. Consider $0\leq s\leq t$. What is the correlation between $W(t)$ and $dW(s)$? Is it zero? Or is it $dt$, which is almost zero? Or maybe this is not a precise question? ...
0
votes
0answers
26 views

How to prove this continuous martingale converges?

Suppose $B = (B_t, t \geq 0)$ is standard Brownian motion. Let $M^\lambda_t := \exp(\lambda B_t - \frac{\lambda^2 t}{2})$ (I have previously shown that this is a martingale). How do I prove that $$ ...
0
votes
0answers
6 views

How can we proove that it's a Gaussian system?

$(W_1, W_2)$ are 2 independent Wiener processes and $$B_1= W_1, ~~~ B_2 = a W_1 + \sqrt{1-a^2} W_2,$$ where $a=(a(t, \omega))_t>0$ and is $(F_t=F_t^{(W_1,W_2)})$-measurable. $0<a<1$. It ...
0
votes
0answers
25 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} ...
0
votes
1answer
29 views

$tB_t$ Integral representation, Brownian Motion

I never learned stochastic differential equations, and so am trying to do some self study. I've arrive at this question: $tB_t\sim N(0,t^3)$? $B_t$ is standard brownian motion. $B_t\sim N(0,t)$, so ...
1
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0answers
10 views

Reflection principle for the modulus of the Brownian Motion

I have the following question. Suppose we define $M(t)=\sup_{0\le s\le t}|B(s)|$, where $B$ is an ordinary Brownian motion in $\mathbb{R}$. How can we compute $P(M(t)\ge a)$? Is it $2P(|B(t)|\ge a)$? ...
0
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0answers
15 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 ...
1
vote
0answers
16 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 ...
1
vote
0answers
23 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 ...
2
votes
1answer
23 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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votes
0answers
26 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 ...
0
votes
0answers
25 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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0answers
20 views

Stochastic calculus

For $l=1,2......$ prove that $E[W^{2l} (t)]=$ $\frac{(2l)!}{2^l l!}$ and $E[W^{2l+1} (t)]=0$ I am trying to find the ways of solving the task from Stochastic calculus, but it seems to be very ...
1
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0answers
27 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 ...
1
vote
1answer
24 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?
1
vote
2answers
29 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)) - ...
2
votes
0answers
15 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 ...
1
vote
0answers
20 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 ...
3
votes
1answer
25 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 ...
1
vote
1answer
36 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 $R = \inf\{t > 1 : B_t = 0\}$. ...
1
vote
0answers
19 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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votes
0answers
26 views

Itō formula for a scalar valued function of the solution of a scalar Itō SODE

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $B$ be a real-valued $\mathcal F$-Brownian motion on $(\Omega,\mathcal ...
7
votes
0answers
104 views

How to calculate the PSD of a stochastic process

Say we have a stochastic process described by a stochastic differential equation (in the Itô sense), and maybe we are able to find an explicit solution of it in terms of deterministic and Itô ...
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0answers
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 ...
2
votes
1answer
445 views

Characteristic function of vector-valued random variables

I just begins my self-study on Brownian motion. I got stuck on the part about random-vector and characteristic function. Here are my questions: I'm not quite get about how characteristic function of ...
2
votes
1answer
34 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\} = ...
3
votes
0answers
38 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: $$ ...
0
votes
0answers
26 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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0answers
12 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 ...
0
votes
1answer
15 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 ...
4
votes
1answer
3k views

Expectation of Stopping Time w.r.t a Brownian Motion

How do you take the expectation of a stopping time with respect to a Brownian motion? The specific question is: $$ \tau = \inf\{ t \ge 0: B(t) \in \{-a, b\}\} $$ I understand the optional stopping ...
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votes
0answers
10 views

Find SDE coefficients such that solution follow given distribution

Consider a cdf F and suppsoe $F^{-1}$ is also known. Denote by W a standard Brownian Motion. 1) Find function G such that random variable $G(W_1)$ has cdf F. Answer: $G(x) = F^{-1}(Ф(x))$, where Ф ...
0
votes
0answers
12 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 ...
0
votes
1answer
26 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, ...
0
votes
1answer
29 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 ...