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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54 views

Brownian motion with Lévy’s Characterization

I want to show that: if for all $\lambda \in \mathbb{R}$ the process $(exp(\lambda X_t-\frac{\lambda ^2}{2}t))_{t\geq0}$ is a $\mathcal{F}^X$ local martingale, then the $\mathbb{R}$-valued process X ...
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1answer
13 views

Strong Markov Property Brownian Motion for Non-Stopping Time

Let $B$ be a Brownian motion and let $\mathcal{F}^B$ be its natural filtration. Define the random variable $$ \tau = \inf\{ t \ge 0 \mid B_t = \sup_{0 \le s \le 1} B_s \}.$$ Now, $\tau$ is not an ...
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0answers
13 views

Difference of Running Maximum of a Reflected Brownian Motion and the Reflected Brownian Motion

For a Brownian Motion $W_t$ and $M_t=\sup_{s<t} W_s$, we know $M_t-W_t$ is a reflected Brownian Motion. For a reflected Brownian motion $X_t=|W_t|$ and the running maximum $M'_t=\sup_{s<t} X_s$, ...
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0answers
16 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 ...
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1answer
17 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 ...
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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} ...
3
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0answers
43 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 ...
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1answer
43 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 ...
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0answers
39 views

Some Kind of Generalized Brownian Bridge

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 ...
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1answer
39 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
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1answer
41 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-dimensional Brownian Motion). I used Ito's formula and got $e^{B_t^1}\cos(B_t^2)=1+\int_0^t ...
3
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0answers
20 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
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0answers
26 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 ...
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0answers
13 views

What is skew-product decomposition?

What is skew-product decomposition of Brownian motions referring to this paper Pauwels ,Rogers
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39 views

How to prove this continuous martingale converges? [on hold]

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 $$ ...
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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 ...
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1answer
37 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 ...
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0answers
13 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 $2P(|B(t)|\ge a)$? ...
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0answers
17 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 ...
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0answers
20 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 ...
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0answers
30 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 ...
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1answer
25 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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27 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 ...
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0answers
29 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 ...
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0answers
26 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 ...
2
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0answers
18 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 ...
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31 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 ...
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1answer
33 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?
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0answers
21 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 ...
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2answers
35 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)) - ...
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1answer
39 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 $R = \inf\{t > 1 : B_t = 0\}$. ...
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0answers
25 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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0answers
84 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 ...
3
votes
1answer
30 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 ...
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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 ...
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0answers
29 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} ...
2
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1answer
35 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\} = ...
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0answers
40 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: $$ ...
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0answers
29 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
20 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 ...
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1answer
18 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 ...
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0answers
14 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 ...
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0answers
22 views

Intuition behind “stochastic orthogonality”

Whilst doing an exercise on the Brownian Motion on a sphere I came across this identity: $$ \langle Z\times B,Z\times B\rangle = 2|Z|^2dt $$ where $\times$ denotes the cross product and $Z$ is a ...
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1answer
33 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 ...
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1answer
28 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, ...
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1answer
16 views

Einstein's number of particles that experienced a certain shift explanation

I am reading a Gardiner's Stochastic Methods handbook and I am wondering about the meaning of the following (this is the very beginning of the chapter): $dn = n \phi(\Delta) d \Delta$ This is ...
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0answers
32 views

Proof of normal distribution property used in Levy's construction of the brownian motion?

I have been trying to follow the construction of Brownian motion by Levy. I need a property about the conditional distribution of the Brownian process in an interior point of an interval given its ...
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0answers
29 views

Can we apply an Itō formula to the solution of a SPDE?

Let $V\subset H\subset V^\ast$ be a Gelfand triple $(\Omega,\mathcal A,\operatorname P)$ be a probability space and $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $(W_t)_{t\ge 0}$ be a ...
2
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0answers
24 views

Long term behavior of Brownian Motion

Let $(B_t)_{t \geq 0}$ be a Brownian motion. The objective is to prove that \begin{align*} \limsup_{t \to \infty} \frac{B_t}{\sqrt{t}} = \infty. \end{align*} By the scaling property of Brownian ...
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1answer
32 views

Why does the Borel-Cantelli lemma finish the job? - Law of Large Numbers Brownian Motion

The objective is to prove that \begin{align*} \text{$\lim_{t \to \infty} \frac{B_t}{t} =0 \qquad$ a.s.} \end{align*} By the strong Law of Large Numbers, we have that: \begin{align*} \text{$\lim_{t ...