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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3
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0answers
36 views

Hitting times in two-dimensional case: expectation of Brownian motion at a hitting time

Consider two Brownian motions $$X_{1t}=\mu t+\sigma_1B_{1t}$$ and $$X_{2t}=\mu t+\sigma_2B_{2t}.$$ Here $B_{1t}$ and $B_{2t}$ are uncorrelated. Let $\tau_1$ and $\tau_2$ be the stopping times: \begin{...
21
votes
3answers
2k 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 ...
1
vote
0answers
18 views

Green's function and strong Markov property for stopped Brownian motion

Let $X(t)$ be a Brownian motion in $\mathbb{R}^n$, stopped at some fixed time $T$. Is there a notion of Green's function for such a Brownian motion? I am guessing that there is, and $G(x, y) : = \...
0
votes
2answers
24 views

Brownian Motion-Independence of Increments

Consider a Brownian Motion $B(t)$ with $B(0)=0$. Suppose $s<t$. I read in a book that while $B(t)-B(s)$ is independent of the past, $2B(t)-B(s)$ or $B(t) - 2B(s)$ is not. Why is this the case? ...
0
votes
0answers
11 views

Local time accumulated on an interval

On Wikipedia, the definition of local time is $$L^x(t) = \int_0^t \delta(x - B_s) ds$$ where $B_s$ is a real-valued diffusion process, and $\delta$ is the Dirac delta function. My question is, are ...
0
votes
0answers
39 views

Brownian motion hitting probability and Martin capacity

Consider a Brownian motion $B_t$ in $\mathbb{R}^n, n\geq 3$ and the ball $B(0, r)$ of radius $r$ around the origin. Let $\overline{C}$ be a compact set inside $B(0, r)$ such that $C$ is open in $B(0, ...
-1
votes
0answers
15 views

Transient Brownian motion and stopping time

Let $B(t)$ be a Brownian motion in $\mathbb{R}^n$, or on a compact Riemannian manifold $M$ of dimension $n$, $n \geq 2$. Let us consider the stopped Brownian motion at a deterministic time $T$ (in ...
1
vote
1answer
53 views

Distributional equality

Let $(W_t)_{t\geq0}$ be a standard Brownian motion. I have to show that the following equality holds in distribution. Does someone has a good hint to show this? $\sup_{t \geq 0}( |W_t| -t) = \sup_{t \...
1
vote
1answer
32 views

a conundrum regarding integrated Brownian motion and fractals

Let $X(t)$ be a Brownian motion. I know that the integral \begin{equation} Y(t) = \int_0^t d\tau ~ X(\tau) \end{equation} is well-defined, since Brownian motion $X(\tau)$ is a.s. continuous. Thinking ...
-1
votes
0answers
40 views

Possible master thesis [closed]

I am searching a topic for my master thesis. My interests are especially probability theory (something with brownian motion would be nice) and fourier analysis (also in an abstract Hilbert space ...
1
vote
1answer
25 views

For a one-dimensional Brownian motion $B_t$ $Ee^{\zeta B_t}=e^{t\zeta ^2/2}\; \text{for all} \; \zeta \in \mathbb{C}.$

A one-dimensional Brownian motion $B_t$ has exponential moments of all orders, i.e. $$Ee^{\zeta B_t}=e^{t\zeta ^2/2}\; \text{for all} \; \zeta \in \mathbb{C}. (2.6)$$ This is given as a corollary to ...
0
votes
0answers
96 views

brownian bridge and supremum

I want to show that: $$ \sup_{u \geq 0} \frac{1}{u} \left( | B_u | - 1 \right) = \sup_{u \geq 0} \left( |B_u| - u \right) = \sup_{0 \leq u \leq 1 } b_u^2 $$ in distribution; with $ B_u = (1+u)b_{\frac{...
4
votes
2answers
920 views

Brownian bridge

Let $W = (W_t;F_t)$, $t \leq 0$ be a standard Wiener process, and let $(X_t)_{0 \leq t \leq 1}$ satisfy the stochastic differential equation $$ dX_t =- \frac{X_t}{1-t}dt+dW_t,\quad 0 \leq t \leq 1,\...
2
votes
1answer
466 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 ...
0
votes
1answer
42 views

brownian bridge definition [closed]

I am trying to solve an exercise and I have trouble with the definition of brownian bridge. "Let (bu , 0 ≤ u ≤ 1) be the Brownian bridge derived by conditioning a one-dimensional Brownian motion (Bu ,...
2
votes
1answer
27 views

Sharpen Doob's Maximal Inequality

Let $B_t$ be a Brownian motion, $B_T^* = \sup_{0\leq t \leq T} B_t$ and $\lambda > 0$. Applying Doob's maximal inequality gives: \begin{align} P(B_T^* \geq \lambda)\leq \frac{\mathbb{E}[B_T^p]}{\...
0
votes
0answers
30 views

Time-changed Brownian Motion

Let $B_t$ be a standard Brownian motion and let $\tau_{-1}= \inf \{ t \geq 0: B_t(\omega) = -1\}$. By the Continuous Time Stopping Theorem we know that \begin{align} Z_t = B_{t \wedge \tau_{-1}} \...
0
votes
1answer
50 views

Correlation between stochastic processes

Question: if $W(t)$ is a standard Brownian motion with $W(0)=0$, what is the linear coefficient between the stochastic processes $W(t)$ and $I(t)=\int_0^t W(s)ds$? I argued as follows: what we want ...
2
votes
1answer
185 views

Conditional expectation brownian motion

Somebody has an idea on how to tackle this quantity $$ \mathbb{E}\left[ \left. \frac{\int_{0}^{T}{{{e}^{\alpha {{W}_{t}}}}}dt}{\int_{0}^{T}{{{e}^{-\alpha {{W}_{t}}}}}dt+\int_{0}^{T}{{{e}^{\alpha {{W}_{...
0
votes
0answers
27 views

Independence between the first exit time from an interval and the value of Brownian motion at this first exit time

Suppose you have an arithmetic Brownian motion (or Brownian motion with drift ) called X, started at a level x such that a < x < b, where a and b are two real points . Define tau as the first ...
2
votes
1answer
90 views

Lebesgue Thorn and Brownian motion

Let $S$ be the unit sphere in $\mathbb{R}^3$ and $\Theta=\{(x,y,z), x\geq 0, z^2+y^2 \leq \frac{1}{10}e^{-1/x^2}\}$. Try to show that $\exists \delta>0,\forall x\in (-1,0)$,$$ P^x(W_t \text{ hit ...
0
votes
0answers
17 views

Given a set of 1-dimensional Brownian motions $\{a_i\}_{1\le i\le K}$, what is the average hitting time between $a_1$ and $\{a_i\}_{2\le i\le K}$?

Given $K$ Brownian motions $\{a_i(t), t\ge 0\}_{1\le i\le K}$ contained within interval $[0,s]$, the boundaries at 0 and s are reflected. Assume at time $t=0$, the initial locations are $a_1(0)\le a_2(...
2
votes
1answer
197 views

Long Range Dependence, Fractional Brownian Motion

A stationary sequence $(X_n)_{n\in\mathbb{N}}$ exhibits long-range dependence if the autocovariance function $\rho(n):=\mathrm{cov}(X_k,X_{k+n})$ satisfy $$\lim\limits_{n\to\infty}{\rho(n) \over cn^{-\...
2
votes
0answers
51 views

Sum of Wiener, limit in probablitity

Show that the sequence is convergence in probability and set the limit of it: $$\sum\limits_{k=n}^{2n-1}\left(W_{(k+1)/n}^2-W_{k/n}^2-\frac{1}{n}\right)\left(W_{(k+1)/n}-W_{k/n}\right).$$ If there ...
0
votes
1answer
156 views

Sojourn time for Brownian motion with drift

Assume that $X(t) = \mu t + \sigma W(t)$, where $W$ is a standard Brownian motion and $\mu > 0$, and define $$ T = \int_0^\infty I(X(t) \in (0,\delta))\ dt $$ where $\delta >0$. What is $...
0
votes
0answers
32 views

System of Stochastic Differential Equations (SDEs) from Diffusion on Manifold

I am looking at a system of SDEs due to Brownian motion on a 3d Riemannian manifold (see e.g. Ito, 1962, The Brownian Motion and Tensor Fields on Riemannian manifolds). I have reduced the associated ...
1
vote
1answer
70 views

Compute $\int_1^2 B_t \; dB_t$

I have to compute the following Ito integral: $$\int_1^2 B_t \; dB_t$$ where $(B_t)_{t \geq 0}$ is the 1-dimensional Brownian Motion. In the definition of Ito integral, the integral is taken from $0$ ...
0
votes
0answers
17 views

For showing measurability of Brownian motion, how does this set equality holds?

It is stated that the the following set equality easily comes from continuity of paths of Brownian motion $B_t$, but I can't seem to make sense of it - $$\{(\omega,t)\in \Omega\times (0,\infty) : ...
1
vote
0answers
11 views

On the conditional distribution of $B_{(s+t)/2}$ conditionally on $(B_t,B_s)$, for Brownian motion $B$

I've been reading stuff about Brownian motions and all that, and I came across the following statement: On proving that $B_{\frac{s+t}{2}}\sim N(\frac{x+y}{2},\frac{t-s}{4})$ conditionally on $B_s=x,...
2
votes
0answers
40 views

martingale square integrable

Let $X_t=\int_0^te^{W_s}dW_s$ and $Y_t=\int_0^tW_sdX_s$. How to show that $X$ and $Y$ are martingale square integrable? ($W_t$ - Wiener) It it enough to show that $\mathbb{E}X_t^2<\infty$, $\...
1
vote
2answers
20 views

(locally) square integrable process

We are given a process $\left(X_t\right)_{t\geq 0} = \left(e^{aW_t^2}\right)_{t\ge0}$, where $W_t$ is Wiener process, $a > 0$. Check for which $a$: 1) $\mathbb{E}\int_{0}^{\infty} X_s^2 ds <\...
1
vote
1answer
80 views

I've found two different definitions of a cylindrical Brownian motion and don't understand why they are consistent

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space $\left(...
5
votes
1answer
2k views

How to derive the Ornstein-Uhlenbeck Stochastic Integral Equation?

I have a question regarding the Ornstein -Uhlenbeck process. We have a simplified version with Stochastic Integral Equation: $X_t=-a\int^t_0 X_s\,ds +B_t$. B is the Brownian motion. And its analytic ...
3
votes
1answer
44 views

Stochastic differential equation substitution reasoning?

I need help to explain reasoning behind why we choose certain substitutions to solve SDE. After choosing the correct substitution the solution of the SDE are often quite easy. However I have trouble ...
0
votes
0answers
31 views

Changing the order of integration for Brownian motion (outer integration over the range of inner integration)

$X_t$ is bounded Brownian motion and it can be even standard Brownian motion if you wish. I want to express $E[\int_{0}^{T}\int_{0}^{t}X^{n}dsdt]$ as a function of $E[\int_{0}^{T}X^{n}dt]$ For ...
2
votes
1answer
104 views

Show that $\mathbb{P}(B_t<x,B_s<x)\geq\mathbb{P}(B_t<x)\mathbb{P}(B_s<x)$ for $0<s<t$, $x>0$, and $B$ Brownian motion

Let $B$ be a standard Brownian motion. How can we show that $$\mathbb{P}(B_t<x,B_s<x)\geq\mathbb{P}(B_t<x)\mathbb{P}(B_s<x)$$ for $0<s<t$ and $x>0$ without actually computing the ...
1
vote
2answers
33 views

What is the distribution of the subtract of two random variables?

Definition) A stochastic process $\{X(t), t \geqslant 0\}$ is said to be Brownian motion process with drift coefficient $\mu$ and variance parameter $\sigma^2$, if it satisfies that $X(0)=0$. $\{X(t)...
1
vote
1answer
36 views

Compute expectation and covariance of Brownian bridge

Let $\{X(t), t \geqslant 0\}$ be a standard Brownian motion. That is, for every $t \gt 0$, $X(t)$ is normally distributed with mean $0$ and variance $t$. Then $\{X(t), 0 \leqslant t \leqslant 1 | X(1)...
0
votes
0answers
22 views

Why are Brownian Motion and Levy processes beginning “almost surely” at 0?

I am studying stochastic calculus, and I had a question about the definition of both Brownian motion, as well as Levy processes. So the formulation that I have seen both on Wikipedia and my textbook(...
1
vote
0answers
12 views

Please verify the solution about Brownian motion process.

Problem Let $Y(t)$ denote the amount of time by which the racer is ahead when $100t$ percent of the race has been completed. $\{Y(t), 0 \leqslant t \leqslant 1\}$ is modeled as a Brownian motion ...
2
votes
1answer
42 views

Solving Langevin equation

In a past exam paper that I am looking at, there is the following question: Given that the displacement, $\mathbf{x}$, of a particle in $3$-dimensional Brownian motion is given by: $$m\ddot{\...
1
vote
1answer
37 views

Verifying a Brownian motion through the Laplace transform

Let $X(t)$ be a continuous stochastic process and $\mathcal G(t)$ be the $\sigma$-algebra generated by $\{X(\tau) : \tau\leq t \}$. Suppose that for any $0\leq s\leq t$ and $\lambda\in\mathbb C$ ...
1
vote
1answer
28 views

From brownian bridge to brownian motion proof

Let $B_t$ be a brownian motion. and let $\{W_t=B_t-tB_1:0\le t\le 1\}$ be a brownian bridge. Now let $Y_t=(1+t)W_{t\over 1+t}$. Proof that $Y_t$ is a brownian motion in $[0, \infty)$ My attempt: 1) $...
2
votes
1answer
47 views

Is Brownian motion on $[0,b]$ bounded?

Is Brownian motion on $[0,b]$ bounded? Or at least bounded with probability one. Since Brownian motion is continuous with probability $1$, I guess the answer is YES.
1
vote
0answers
25 views

Is $B_{t\wedge H_a}$ bounded in $L^2$?

Let $a >0$, $(B_t)_{t\geq0}$ be a standard Brownian motion. Define the stopping time $$H_a := \inf\{t \geq 0 : B_t \geq a\}.$$ Then is the martingale $M_t$ where $M_t: = B_{t\wedge H_a}$ bounded ...
1
vote
1answer
173 views

Verifying a standard Brownian Motion? [closed]

Let $\{X_t, t\ge 0\}$ be a standard Brownian motion process. For a fixed positive number s and all $t\ge 0$, we define $Y_t = X_{t+s} - X_s$. Is $\{Y_t, t\ge0\}$ a standard Brownian motion? Attempt: ...
1
vote
0answers
23 views

Brownian Motion maximum process intuition

I am studying the maximum value of a Brownian Motion (BM) on an interval of time (as explained here between boxes 28 and 40) and I am having an issue aligning intuition with the mathematical result. ...
1
vote
0answers
29 views

Brownian motion independent RVs

Let $(W_t)_{t\in\lbrack 0,T\rbrack}$ be a standard Brownian motion. Does there hold that $W_s(W_t-W_s)$ and $W_k(W_l-W_k)$ for $0\leq s<t\leq k<l\leq T$ are independent RVs?
-2
votes
1answer
30 views

What is the difference between these two formulas that price a stock? [closed]

What is the difference between these two formulas? They are both related to the price of a stock in the black-scholes model. The fact that the second one uses $t$ as a subscript which means it's not a ...
3
votes
2answers
84 views

Crossing of Brownian Motion Sample Paths

I would like to ask for a more rigorous statement and proof of Lemma on page 5 of this paper. In essence, it states that two distinct sample paths of a Brownian motion does not strictly cross (meaning ...