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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Intuition Behind this Theorem About Brownian Motion

I am having a hard time with the intuition behind some of the representation theorems dealing with Brownian Motion. I think if someone can simply explain the intuition behind this theorem then ...
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17 views

Expectation $ \mathbb E^{\mathbb P^{\tau}}_t\left[ X_{\tau}(T)Y_{\tau}\right] $ and measure change.

Under a probability space $(\Omega, (\mathcal F_t), \mathbb P)$ consider two processes $X$ and $Y$ following the SDEs $$ \frac{dX_t}{X_t} = \mu_t \ dt + \sigma(t,T) \ dW^1_t $$ $$ \frac{dY_t}{Y_t} = ...
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12 views

Equivalence of different definitions of (Laplace) Green function

Fix an open set $D \subset \mathbb{R}^d$. Usually the (Laplace) Green function is defined as the solution to the boundary value problem $$ \begin{cases} \Delta u(x) = \delta_y(x) \quad x \in D\,, \\u(...
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Finding g$^∗$, τ$^∗$ in 1-dimensional Brownian motion

How do I find g$^∗$, τ$^∗$ such that g$^∗$(s, x) = sup$_τ$E$^{(s,x)}$[e$^{−ρ(s+τ)}$B$_τ$$^2$] = E$^{(s,x)}$[e$^{−ρ(s+τ^{*})}$B$_{τ^*}^2$ ] , where B$_t$ is 1-dimensional Brownian motion, ρ > 0 is ...
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55 views

Can we show $\int_0^tf(s){\rm d}B_s=-\int_0^tf'(s)B_s{\rm d}s$ for $f\in C^1(\mathbb R)$?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(B_t)_{t\ge 0}$ be a Brownian motion on $(\Omega,\mathcal A,\operatorname P)$ and $f\in C^1(\mathbb R_{\ge 0})$. Can we show that $$\...
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40 views

How can we solve $\frac{{\rm d}^2u}{{\rm d}t^2}(t)=-c^2\lambda u(t)+\varepsilon\sqrt{\lambda}\frac{{\rm d}B}{{\rm d}t}(t)$?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(B_t)_{t\ge 0}$ be a Brownian motion on $(\Omega,\mathcal A,\operatorname P)$ How can we solve $$\frac{{\rm d}^2u}{{\rm d}t^2}(...
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+100

Brownian Motion in Confined space, any results?

I am searching for work regarding Brownian motion in a confined space, like a sphere or a cylinder, where the wall will serve as reflection boundary. I am wondering if it is possible to derive results ...
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22 views

Is there a way to maximize this probability by taking the derivative of the cumulative normal distribution function?

I'm self-studying Brownian motion and encountered the following problem. I understand the author's solution, and it is clear why maximizing the right-hand side of the inequality provides such $t$ ...
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14 views

Brownian Bridge Probability

Doing a project I have found in some papers that the (discretised) probability of the Brownian Bridge (which has $S(n)$ as initial value and $S(n+1)$ as final value, where S follows a Geometric ...
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474 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 ...
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12 views

Stochastic wave equation

Let $D:=(0,a)$ for some $a>0$ $H:=L^2(D)$, $$e_n(x):=\sqrt{\frac 2a}\sin\frac{n\pi x}a\;\;\;\text{for }n\in\mathbb N\text{ and }x\in D$$ and $$\lambda_n:=\left(\frac{n\pi}a\right)^2\;\;\;\text{...
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1answer
47 views

Expectation of the first passage time of $T_{a,b}$ [duplicate]

Let $(\Omega, \mathcal{F},\mathbb{P})$ be a probability space and let $W_t$ be standard Wiener process and $$T_{a,b}=\inf\{t:W_t=a+bt\}$$ where $a$ and $b$ are costant.I want to get expectation of $...
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2answers
59 views

Correlated brownian motions and Lévy's theorem

$W^{(1)}_t$ and $W^{(2)}_t$ are two independent Brownian motions. How can I use Lévy's Theorem to show that $$W_t:=\rho W^{(1)}_t+\sqrt{(1-\rho^2)} W^{(2)}_t,$$ is also a Brownian motion for a given ...
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0answers
11 views

Can Fluctuation-Dissipation Theorem Apply to Magnetic Forces in Multi-Spin Systems [closed]

Let's say I have multiple spin systems (atoms in a protein) in a solution of water and the spin systems are all producing a magnetic field $\mathrm{B_{loc}}$ that affects nearby spin systems. Will the ...
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6 views

Proof of Exponential Decay Pattern of Time Correlation Functions for $\mathrm{B_{loc}(t)}$ in NMR Spectroscopy

For a given protein, I know that the NMR Spectroscopy magnet generates a field $\mathrm{B_o}$ and that the interactions with the spins in the local environment generates a much smaller field $\mathrm{...
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23 views

Integrated Brownian Bridge is a Gaussian Process

Let $W(t),t \in [0,1]$ be a (Standard) Wiener Process. The Brownian Bridge $B(t), t \in [0,1]$ can be constructed via $B(t):=W(t) - t \cdot W(1)$ and is a Gaussian process with zero mean and ...
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1answer
48 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 ...
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25 views

limit of the absolute value of brownian motion

i'm trying to figure out if the limit of the absolute value of a brownian motion goes to $\infty$ as t goes to $\infty$. from the law of iterated logarithm i know that $\limsup_{t\to\infty} \frac{B(...
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25 views

Distribution of marginal Wiener process.

Let $(W^1,W^2)$ be a two-dimensional Wiener process with correlation $\rho$. Let $\mathcal{F}^1_t$ be the filtration generated by $W^1$ up to $t$. I would intuitively think that for $h$ measurable ...
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35 views

$(B_t)_{t\ge 0}$ be Brownian motion. Then $\xi \mapsto \mathbb E e^{i\xi B_t}$ is an analytic function. [closed]

Let $(B_t)_{t\ge 0}$ be a one-dimensional Brownian motion. Then $\xi \mapsto \mathbb E e^{i\xi B_t} \; \text{for all} \; t\ge 0, \xi \in \mathbb{R}$ is an analytic function. A more general question ...
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1answer
59 views

Expectation of Product of Ito Integrals wrt Independent Brownian Motions

Let $W_1(t)$, $W_2(t)$, $W_3(t)$ be independent Brownian motions and $f$, $g$ smooth functions. I want to know if the following is true: $$ \mathbb{E}\left[ \left( \int\limits_0^t f(...
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1answer
77 views

Behavior of fundamental solution to heat equation after projection

I am considering the behavior of $$\frac{1}{h}\|(1-P_h)S(h)v\|,\tag{1}$$ and $$\frac{1}{h}\|(1-P_h)S(h)P_hv\|,\tag{2}$$ as $h\to 0^+$ for a fixed good enough $v$. I hope to show one of them converges ...
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1answer
50 views

Reference: Computing Martin Capacity

For Borel set $A$ the Martin Capacity is defined as: $\mathrm{Cap}_{M}(A)=[\inf\{\int \int \frac{G(x,y)}{G(0,y)}d\mu(x)d\mu(y):\mu \mbox{ probability measure on }A \}]^{-1}$ and Green's function $G(x,...
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37 views

Why is a continuous Lévy process twice integrable?

In his textbook "Wahrscheinlichkeitstheorie" (de Gruyter, 2008), Jochen Wengenroth shows (p. 144) that if $(X_t)_{t\in[0,\infty)}$ is a continuous, real-valued Lévy process with $X_t\in \mathcal{L}_2$ ...
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1answer
148 views

Given $\mathbb Q$ and $X_t$ is $\mathbb Q$-Brownian, find $\frac{d\mathbb Q}{d\mathbb P}$ / Uniqueness of Brownian or Radon-Nikodym derivative

The problem: Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t \...
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Brownian motion, modifications vs indistinguishablity

In Protter's book Stochastic Integration and Differential Equations and in uncountable other sources, they mention the continuous sample paths of the brownian motion. That is: It holds that $$t\...
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1answer
57 views

Confusion about the average distance traveled on a $1$D random walk

The average absolute distance on a one dimensional random walk is supposed to be $\sqrt{n}$. Where $n$ steps are taken from the origin or $n$ is the time. I don't have an intuitive understanding or ...
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6 views

What is the rate of convergence of Brownian motions Increments?

Would like to know what the rate of convergence of brownian motion is? I know each brownian motion increment is distributed with N(0,t) so do i need to apply a CLT?
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What does it mean that a stochastic process is independant of a filtration?

Let $(\Omega ,\mathcal F,P)$ a proba space and $(\mathcal F_t)_{t\geq 0}$ a filtration. Let $(B_t)_t$ a Brownian motion adapted to $\mathcal F_t$. We know that $(B_{t+s}-B_t)_{s}$ is independent of $\...
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0answers
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What is the distribution of the maximum of scaled Gaussian random walk?

A Gaussian random walk is the sum of standard normal variables $$Z(n) = \sum_{i=1}^n X_i,$$ where $X_i\sim N(0,1)$. What I mean by a scaled Gaussian random walk is the following: $$ U(n) = \frac{Z(n)}{...
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1answer
501 views

Expectation of Exponential of Stochastic Integral

Let $z$ be the standard Brownian motion, $\omega$ an element of the sample space. Is it true that $$ \mathbf E\bigg[\exp\Big(\int_0^t f(\omega,s)\,\mathrm dz(s)\Big)\bigg] = \mathbf E\bigg[\exp\Big(\...
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Expected value of the exponential of a Geometric Brownian motion

I am trying to compute the following expectation: $$ E[ \exp (A_T)], $$ where $A_T = - C \int_{0}^{T} \exp( 2 \alpha W_t - \alpha^2 t) dt $, with $C$ and $\alpha$ positive constants, $W_t$ a standard ...
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27 views

Black scholes model for down and out European call option using Monte Carlo

I tried to implement Matlab program computing the price of the European down and out call option using Monte Carlo and Euler discretization scheme. I have initial price S0=50, strike K=50, barrier ...
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1answer
33 views

Hitting time or normalized Brownian motion (divided by square root of t)

What I mean by a normalized Brownian motion is the following: $$ U(t) = \frac{W(t)}{\sqrt{t}}, $$ where W(t) is a standard Brownian motion (with $\mu = 0$ and $\sigma = 1$). Is there a name for such a ...
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1answer
19 views

Identity of Running maximum

let $X_t$ denotes a arithmetic Brownian motion process. I am wondering if the following identity is true ? $$ \mathrm{P}\left[\sup_{0 \le s \le t} \mathrm{e}^{X_t} < x\right] = \mathrm{P}\left[\...
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Why is a Brownian Motion completly specified by its increments?

Without any formal base frame I want to know why a Brownian Motion is completly specified by its increments? Even though I think here is no need to give a Reference you can see the following. ...
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1answer
98 views

quadratic variation of brownian motion doesn't converge almost surely

I just came across the following remark: If $(B_t)_{t\geq0}$ is a one dimensional Brownian motion and if we have a subdivison $0=t_0^n<...<t_{k_n}^n=t$ such that $\sup_{1\leq i\leq k_n}(t_i^n-t_{...
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Probability that Brownian motion falls between two piecewise constant functions

I'll first present the problem, and then describe my motivation: Suppose $a_j \in \mathbb{R}$, $b_j \ge 0$, and $0 = t_0 < t_1 < \cdots < t_J$ are time points. Let $W_t$ be a standard ...
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2answers
61 views

Is there a boundary in probability for Brownian motion?

For a standard Brownian motion $W_t$ and a given crossing probability $\alpha < 1$, I want to have a boundary function $f(t) > 0$, such that the probability that $W_t$ ever crosses the boundary ...
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1answer
32 views

Continuity of the probability that a Brownian motion with drift hits an upper barrier before the lower barrier in the drift

Let $W$ be a Brownian motion and $u:\mathbb{R}_+ \to \mathbb{R}_+$ an upper barrier and $l:\mathbb{R}_+ \to \mathbb{R}_-$ a lower barrier. Let $$\tau_u(\mu) = \inf\{ t \colon \mu t + W_t \geq u(t)\}$$...
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24 views

Marginal mean of two dimensional Brownian motion.

Let $(B^1,B^2)$ be a two-dimensional Brownian motion. Let $t>s$. Is it true that $$ E[B^1(t) \lvert B^2(t),B^2(s) ] = E[B^1(t) \lvert B^2(t),B^2(t)-B^2(s) ] = E[B^1(t) \lvert B^2(t)] $$ since $(x,...
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63 views

Family of partitions, s.t. the quadratic variation of a BM diverges a.s.

This question is about a specific step in the solution of exercise 1.13 a) of the book "Brownian Motion" by Peres and Mörters (https://www.stat.berkeley.edu/~peres/bmbook.pdf). The exercise is on page ...
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Brownian motion with Lévy’s Characterization

I want to show that: if for all $\lambda \in \mathbb{R}$ the process $\left(\exp\left(\lambda X_t-\frac{\lambda ^2}{2}t\right)\right)_{t\geq0}$ is a $\mathcal{F}^X$ local martingale, then the $\mathbb{...
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0answers
26 views

Brownian motion hitting probability of boundary and going outside

I was solving an exercise which asks the reader to calculate the probability that a Brownian particle $B(t) = (B_1(t),...,B_n(t))$ starting at the origin in $\mathbb{R}^n$ will strike the surface of a ...
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1answer
118 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 \...
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18 views

Marginal conditional mean of two dimensional Brownian motion, using more than one time point.

EDIT: I found the error! I do not think this question is relevant for anyone, but I cannot find out how to delete it - please feel free to if you have the read this and have the option. I have ...
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1answer
141 views

Evaluating the distribution involving a Brownian motion.

I'm trying to solve one exercise closely related to this question. Since I don't have an answer yet, I thought to post a new question with my thoughts about the problem. I hope this does not break any ...
3
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1answer
33 views

Is the supremum of an almost surely continuous stochastic process measurable?

Let's take a stochastic process $(X_t)_{0\leq t \leq 1}$ and assume that the sample paths are almost surely continuous. Let us define $S \equiv \sup_{t \in [0,1]} X_t$. How can we show that $S$ is ...
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0answers
24 views

What is the time unit of Wiener process?

For a Wiener process, we know that the mean is $0$ and the variance is $\delta t$. But I am not sure how to interpret $\delta t$. For starter, what is the unit of the time? If the Wiener process is ...
0
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1answer
31 views

Question about motion. [closed]

I don't know how to solve this problem . Please anyone help. Question:- An object start moving from immobility with a acceleration of " f m/s^2". At the end of every t seconds it increases it ...