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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A Question on the Scaling Invariance of Brownian Motion

I read the following paragraph. Let $B_t, \ t \in [0, \infty)$ be a standard linear Brownian motion. For each $q > 4$, define the following sequence of sets. $$ \Omega_k := \left\{\omega \in ...
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14 views

Cameron Martin Theorem

I am struggling with two versions of the Cameron Martin Theorem. 1) We define the measure spaces $(\Omega,\mathcal{F},P)$ and $(C[0,1],\mathcal{C},\mathbb{L}_0)$, where $\mathcal{C}:=\sigma(f\mapsto ...
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0answers
18 views

Probability of hitting two balls for $d\geq 3$

The hitting probability for balls centered at origin is $P_{x}(T_{B_{r}(0)}<\infty)=\frac{r^{d-2}}{|x|^{d-2}}>0$ where $|x|>r$. So is it immediate that $$P_{x}[(T_{B_{r}(0)}<\infty) ...
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1answer
11 views

Zero hitting probability for positive measure sets in $\mathbb{R}^{d}$

In $d\geq 3$, we have that BM is transient a.s. i.e. $lim_{t\to \infty}|B_{t}|=\infty$. But does this imply $P_{x}(T_{A}<\infty)=0$ for some type of Borel sets $A\subset \mathbb{R}^{d}$ with ...
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1answer
22 views

Equivalent Stopping Times for Brownian Motions

For standard Brownian motion $B$, define stopping time $T_1:=\inf\{t>0: B_t = 3\}$ and $T_2:=\inf\{t>0: B_t = -3\}$ and $T_3 := \min\{T_1, T_2\}$. Can I say that $T_3 = \inf\{t>0, B_t \in ...
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10 views

Testing numerical solvers with analytic solution to Ornstein-Uhlenbeck SDE?

I have an SDE I want to solve numerically that is fairly close to the Ornstein-Uhlenbeck process: $$ dx_t=θ(μ−x_t)dt+σdW_t $$ which has analytic solution $$ ...
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7 views

Probability that Brownian motion crosses through opposite sides of a sphere

The problem is to find: $P_{x}[(t<T_{B_{0,r}^{+}}<\infty)\cap(t<T_{B_{0,r}^{-}}<\infty)]$, where $B_{0,r}^{+}\subset (\mathbb{R}^{3})^{+}$ i.e. $B_{0,r}^{+}=\{x\in B_{0,r}^{+}: _{3}\geq ...
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1answer
92 views

Brownian Motion inequality (related to Dvoretzky-Erdoes test)

i have the following question: Let $B(t)$ be a d-dimeansional Brownian motion $d\ge 3$, and $f$ be a monoton increasing function from the positive reals to the positive reals. Let $A_n=(\exists t\in ...
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1answer
38 views

Is this Stochastic integral a martingale ?

Let $(B_t)$ be a Brownian motion and set $X_t = \int_0^t B_t^2 dB_s$. Is $X_t$ martingale? My idea is to rewrite $X_t$ in terms of Ito's Formula $(f(x) = \frac{1}{3}x^3)$ $X_t = \int_0^t B_t^2 dB_s ...
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1answer
16 views

Brownian Motion and Progressive Process

Let $B_t$ be a Brownian motion. Define sign function as follows. $sign(0) = 0$ and $sign(x) = \frac{x}{|x|}, \forall x \neq 0$. I do not know how to show the following two questions, especially on the ...
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1answer
19 views

Stopping Time and Brownian Motion [closed]

Let $B_t$ be a Brownian motion. Let $a < 0 < b$. Consider $\tau: = \min\{T_a, T_b\}$ where $T_a := \inf\{s \geq 0: B_s \leq a\}$ and $T_b := \inf\{s \geq 0: B_s \geq b\}$, namely, the first ...
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1answer
26 views

Stochastic integral a Martingale? [closed]

Let $(B_t)$ be a Brownian Motion wrt. to a filtration $(\mathcal{F}_t)$. Set $X_t = \int_0^t B_s d B_s^7.$ Is $X_t$ a Martingale?
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1answer
45 views

How to show the following definition gives Wiener measure

On the first page of Ustunel's lecture notes, he defines the Wiener measure in the following way: Let $W = C_0([0,1]), \omega \in W, t\in [0,1]$, define $W_t(\omega) = \omega(t)$. If we denote by ...
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0answers
7 views

Showing that $P(\sigma<\infty)=e^{-2am}$, where $\sigma=\inf_{t>0}\{B_{t}=mt+a\}$

This is homework so no answers please Any mistakes: We showed that $X_{t}=e^{\lambda B_{t}-\frac{\lambda^{2}}{2}t}$ is a martingale for any $\lambda>0$. So let $\lambda=2m$, then ...
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1answer
31 views

Variance process of stochastic integral and brownian motion

Let $(W_t)$ be a Brownian motion with respect to a filtration $(\mathcal{F}_t)$. For all $t \geq 0 $ set $$X_t = \int_0^t W_s^2 \mathrm{d} W_s,\qquad Y_t = W_t^7.$$ Find the covariance process ...
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2answers
28 views

Brownian and Brackets

A continuous martingale with deterministic bracket must be a Brownian motion. Is this statement ture or not, please? If true, how to show it? If not, what is a counter example?
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1answer
12 views

Martingale property of negative Brownian motion

Let $B_t$ be Brownian motion, with $B_0=0$. Next define $M_t=-B_t$. Have I understood it correctly that $M_t$ is not a Martingale? $E[M_t]=0$ $E[M_{t+1}|M_t]=-M_t$ and therefore not a Martingale? ...
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1answer
31 views

Why is the canonical filtration of a Brownian motion left-continuous?

Let $\{W_t, t\geq 0\}$ be a Brownian motion, and has a.s. continuous sample paths. Let $\{\mathcal{F}^W_t, t\geq 0\}$ be the canonical filtration, i.e. $\mathcal{F}^W_t=\sigma(W_s, 0\leq s\leq t)$. ...
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2answers
34 views

simple stochastic differentiate

someone can help me to differentiate $$a(t-1)+bt+(1-t)\int_{0}^{t}\dfrac{dB_s}{1-s}?$$ I've tried but I really don't know how to do with the last part.. Thank you somuch for your help
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2answers
56 views

Stratonovich integral

I'm having some troubles to calculate the Stratonovich integral $I(sin)(t)=\int_{0}^{t}\sin{B_{s}}dB_{s}$. I've tried with the limit of ...
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1answer
39 views

Conditional Integral of Square of Brownian Motion?

I am struggling to compute the expectation and variance of the following, where $W(s)$ is a standard Brownian motion: $$ X := \int_{0}^{A}W(s)^2ds$$ $$ Y:= \int_0^AW(s)ds $$ $$E[X\mid Y] = \space ?$$ ...
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1answer
23 views

Proving Brownian Motion has Stationary Increments

In Oksendal's 'Stochastic Differential Equations', we define Brownian Motion as follows: Fix $x\in\mathbb{R}^n$ and define for $y\in\mathbb{R}^n$: $$p(t,x,y)=(2\pi ...
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1answer
126 views

The Brownian motion process in Sheldon M. Ross

Today I study Brownian Motion and Geometric Brownian Motion using textbook: An Elementary Introduction to Mathematical Finance, Third Edition by Sheldon M. Ross but I missed the class because I was ...
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1answer
61 views

Solve a linear stochastic differential equation [closed]

I don't know how to find a solution of this stochastic differential equation: $dX_{t}=(1+\delta \mu X_{t})dt+\delta X_{t}dB_{t}$ Where $B_{t}$ is a standard Brownian motion and $\mu$ and $\delta$ ...
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1answer
44 views

Prove $A_t := W_t^3-3t W_t$ a martingale

Given prob space $(\Omega, \mathscr{F}, P)$ and a Wiener process $(W_t)_{t \geq 0}$, define filtration $\mathscr{F}_t = \sigma(W_u : u \leq t)$ Let $(A_t)_{t \geq 0}$ where $A_t = W_t^3 - 3tW_t$. ...
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0answers
19 views

Conditional independence in Markov family. [closed]

Suppose that X , ($\Omega, \mathbb{F}$), ${\lbrace}{P^x}{\rbrace}_{x \in \mathbb{R}^d} $ is a markov family with shift operators ${\lbrace} \theta_s{\rbrace}_{s\geq 0}$. Using the fact that for every ...
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0answers
17 views

How can a Jump Diffusion Process be defined on White Noise rather than Brownian Motion

Jump diffusion processes are a Brownian process with randomly occurring jumps: http://en.wikipedia.org/wiki/Basic_affine_jump_diffusion Can such a process be simulated using white noise rather than ...
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1answer
23 views

Cauchy distribution for Brownian motion

This is homework so no answers please Problem: Find distribution of $(B_{1}(T_{a}),B_{2}(T_{a}))$, where $T_{a}=inf_{t\geq 0}\{B_{2}(t)=a\}$ Any mistakes: $T_{a}=inf_{t\geq 0}\{B_{2}(t)=a\}$ has ...
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1answer
50 views

Random walk and Occupation measure

This is homework so no answers please I want to find for some $A\subset \mathbb{R}$ the limit $$\lim_{n\to \infty}\mu_{n}(A)=\lim_{n\to ...
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1answer
27 views

Help understanding a proof of non-differentiablity of Brownian motion

The following statement and proof are taken from the book Brownian Motion by Peter Morters and Yuval Peres. Since I initially didn't fully understand the proof I added some clarifications and I was ...
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0answers
47 views

Showing $E[e^{-\lambda \tau_{a}\wedge\tau_{-a}}]=sech(a\sqrt{2\lambda})$

This is homework so no answers please. For $\tau_{a}=inf_{t}(B_{t}=a)$ , we already know $E[e^{-\lambda \tau_{a}}]=e^{-\sqrt{2\lambda}a}$. By $B_{t}$ I mean Brownian motion. The question is to show: ...
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1answer
39 views

Covariance of m-fold integrated Wiener process

The problem I'm trying to perform a Bayesian approach to the Maximum Likelihood Estimation procedure of Wecker and Ansley (1983). To this end, I need to compute the full likelihood of the data given ...
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0answers
48 views

Law of a geometric brownian motion first hitting time (proof checking)

I need to use it in a small step in the middle of a simulation and I think I'm not getting correct results to this probabilities and so for my all subsequent simulation. Could someone ...
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2answers
29 views

Conditional expectation of Brownian motion given its absolute value.

Assume that $W_t$ is Brownian motion (1-D) and that $t<T$. How can I compute $$E(W_t||W_T|),$$ the conditional expectation of $W_t$ given $|W_T|$, i.e. with respect to the $\sigma$-algebra $F$ ...
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2answers
40 views

Itô process and covariance of two Brownian motion

I'm a novice in studying the stochastic different equation, and didn't know whether I have describe the question correctly. Here is the question: Suppose $$\begin{array}{rcl} ...
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1answer
25 views

mean hitting time of a level and growth rate of maximum process

Let $X_t$ be the absolute value of Brownian motion starting at $0$, let $\tau_x$ be it's first hitting time of the level $x>0$, and let $M_t$ be it's running maximum up to time $t$. Suppose we knew ...
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0answers
48 views

Expectation of absolute value of Brownian motion

I'm working on this problem that I can't seem to figure out. The problem involves a 1-dimensional Brownian motion, $B_t$, where the subscript denotes the time, and it asks me to show that the ...
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0answers
26 views

Ito's integral from the definition

I am doing Oksendal's book exercises one by one. I got stuck in 3.2. I need to prove, from the definition that $$\int_{0}^{t}B_s^2\text{d}B_s=\frac{B_s^3}{3}-\int_{0}^{t}B_s\text{d}s,$$ where ...
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1answer
22 views

Probability Brownian Motion doesn't hit a point in the limit.

This is a question from Revuz and Yor (exercise 3.18) for which I seem to get a different answer. Show that $\lim_{t \to \infty}\,t^{1/2}\,\mathbb{P}\{B_s\leq1\,\forall\, s\in[0,t]\}=\sqrt{2/\pi}$. ...
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1answer
21 views

Reference for the Construction of Brownian Motion

A common method for constructing Brownian motion is referred to as the Levy construction, the Levy-Ciesielski construction, the Ciesielski construction and sometimes seems to be attributed to Wiener ...
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0answers
10 views

Holder continuity, brwonian motion [duplicate]

Let $B$ stand for a brownian motion on a finite interval $[0,1]$. If i am not wrong, i think that there exists a positive constant $c$, such that almost surely, for h small enough , for all $0< t ...
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0answers
38 views

Deriving Spectral density of White noise from Brownian motion

This is homework so no answers please Here is the problem and my answers (so please tell me if I made any mistakes): I am not asking you to compute the sum at the end, but to tell me if I made any ...
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1answer
25 views

Distribution of hitting position of line by brownian motion.

What is known about the distribution of the hitting position of a line by a 2d brownian motion? I've tried to make some simulations of a 2d brownian motion where every computational step has a ...
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1answer
58 views

Brownian motion - Hölder continuity

Let $B$ stand for a Brownian motion on a finite interval $[0,1]$. If I am not wrong, I think that there exists a positive constant $c$, such that almost surely, for $h$ small enough , for all $0< t ...
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1answer
29 views

$\forall p>0$ we get $\displaystyle \lim_{n\to \infty} n^{a_{p}}\sum _{i=0}^{n}|B_{\frac{i+1}{n}}-B_{\frac{i}{n}}|^{p}=c_{p}$

This is a Homework question, so please do not answer it. Find real constants $a_{p},c_{p}$ s.t. $\forall p>0$ we get $\displaystyle \lim_{n\to \infty} n^{a_{p}}\sum ...
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0answers
18 views

Separation of variables and Fourier transformation

I know there's another question very similar to this argument. In the book "Probabilità e modelli aleatori" of Enzo Orsingher, at pag 134, it shows that the transiction function of an absorbing ...
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23 views

Brownian motion and proximity to a set

Here is the problem: Given two square planes $S_{1},S_{2}\subset \mathbb{R}^{3}$ oppositely positioned to the origin along the x-axis and $S_{1},S_{2}\perp x$-axis. Also, let ...
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0answers
18 views

Hitting time and its distribution

÷I'm reading an italian book about casual process (Probabilità e modelli aleatori of Enzo Orsingher). At pag 105 there's the probability of the stopping time $T_\beta$. $$P\{T_\beta \leq ...
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1answer
12 views

Exclusion-Inclusion principle for Hitting times of two disjoint sets

Consider disjoint sets A, B and Brownian motion $B_{t}$ with $B_{0}\notin A\cup B$. Let $T_{A}:=inf_{t>0}\{B_{t}\in A\}$. Then, do we get $P(T_{A\cup B}<\infty)=P(T_{A}<\infty)+P(T_{ ...
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1answer
34 views

Simple question about the definition of Brownian motion

I have a question concerning the definiton of Brownian motion. Usually (e.g. on Wikipdia) one demands a brownian motion $\lbrace B_t\rbrace_{t\in[0,\infty)}$ to satisfy the following condition: ...