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Questions tagged [brownian-motion]

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 $t-s$.

25
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3answers
18k views

Integral of Brownian motion is Gaussian?

Let $(W_t)$ be a standard Brownian motion, so that $W_t \sim N(0,t)$. I'm trying to show that the random variable defined by $Z_t = \int_0^t W_s \ ds$ is a Gaussian random variable, but have not ...
9
votes
4answers
4k views

Wiener Process $dB^2=dt$

Why is $dB^2=dt$? Every online source I've come across lists this as an exercise or just states it, but why isn't this ever explicitly proved? I know that $dB=\sqrt{dt}Z$, but I don't know what ...
40
votes
3answers
7k 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 ...
19
votes
4answers
14k views

Density of first hitting time of Brownian motion with drift

I just started learning about Brownian motion and I am struggling with this question: Suppose that $X_t = B_t + ct$, where $B$ is a Brownian motion, $c$ is a constant. Set $H_a = \inf \{ t: X_t =a \}$...
10
votes
2answers
7k views

Prove the time inversion formula is brownian motion

Let $B=(B_t)_{t\geq 0}$ be a brownian motion. Show the time inversion formula $\hat{B}=(B_t)_t\geq0$ is a brownian motion, where for $t \geq 0$ we set $\hat{B}=0$ for $t=0$ and $\hat{B}=tB_{1/t}$ for $...
7
votes
1answer
9k 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 ...
13
votes
2answers
3k views

What is “white noise” and how is it related to the Brownian motion?

In the Chapter 1.2 of Stochastic Partial Differential Equations: An Introduction by Wei Liu and Michael Röckner, the authors introduce stochastic partial differential equations by considering ...
7
votes
1answer
2k 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 $\displaystyle H_n(t,x)=\frac{(−t)^n}{n!} e^{x^2/(2t)} \frac{d^n}{dx^n}e^{-x^2/(2t)}$. ...
8
votes
3answers
3k views

Expected value of average of Brownian motion

For a standard one-dimensional Brownian motion $W(t)$, calculate: $$E\bigg[\Big(\frac{1}{T}\int\limits_0^TW_t\, dt\Big)^2\bigg]$$ Note: I am not able to figure out how to approach this problem. All ...
5
votes
1answer
534 views

Why do we unavoidably (or not) use Riemann integral to define Itô integral?

https://en.wikipedia.org/wiki/Itô_calculus Define $$\int_0^tH_tdB_t\equiv \lim_{n\rightarrow\infty}\sum_{i=1}^nH_{t_i}(B_{t_i}-B_{t_{i-1}})$$ But I'm wondering why not defining this using Lebesgue ...
0
votes
1answer
76 views

Brownian Motion problem 22

For a standard Brownian motion $\{W_t, t\geq 0\}$, find $\mathbb{P}(\max_{ t \in [0,1]}|W_t| <x)$. Page 79-80 of Billingsley, P., Convergence of probability measures, New York-London-Sydney-...
17
votes
4answers
4k views

Showing that Brownian motion is bounded with non-zero probability

How do you show, that for every bound $\epsilon$, there is a non-zero probability that the motion is bounded on a finite interval. i.e. $$\mathbb{P} (\sup_{t\in[0,1]} |B(t)| < \epsilon) > 0$$ I ...
16
votes
3answers
5k views

The Laplace transform of the first hitting time of Brownian motion

Let $B_t$ be the standard Brownian motion process, $a > 0$, and let $H_a = \inf \{ t : B_t > a \}$ be a stopping time. I want to show that the Laplace transform of $H_a$ is $$\mathbb{E}[\exp(-\...
7
votes
1answer
1k views

How to show the following process is a local martingale but not a martingale?

I am confronted with the same problem in this thread, which hasn't had a complete answer yet. In the sequel, we denote by $Y^T$ the stopped process $Y^T_t = Y_{T \wedge t}$. Consider the following ...
10
votes
2answers
1k views

Joint moments of Brownian motion

My approach to this SE question uses the following joint moments of Brownian motion. For $n=1,2$ they are obvious and well-known, the others are not terribly hard to work out. Is there a reference ...
9
votes
2answers
580 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 ...
4
votes
1answer
596 views

“Continuity” of stochastic integral wrt Brownian motion

I'd like to prove a nice property of a stochastic integral with respect to Brownian motion. Let $(H_t)_{t\geq0}$ be a progressive and bounded process that is continuous at $0$ and $B$ a standard ...
4
votes
1answer
360 views

(Ito lemma proof): convergence of $\sum_{i=0}^{n-1}f(W(t_{i}))(W(t_{i+1})-W(t_{i}))^{2}.$

The purpose of this question is to complete my personal exposition on the rigorous proof of Ito's lemma. I have consulted more than half a dozen mathematical finance texts and not a single one, for ...
2
votes
2answers
1k views

conditional expectation of brownian motion

Let $(B_t)_{t\geq 0}$ be a standard Brownian motion in $\mathbb{R}^d$. It is intuitive that, for fixed $s<t<u$ $$\mathbb{E}[B_t\mid \sigma(B_s,B_u)]=B_s+\frac{t-s}{u-s}(B_u-B_s).$$ However, I ...
1
vote
1answer
788 views

Probability Brownian motion is positive at two points

Let $0<s<t$ and $(B_r)_r$ is Brownian motion. Does anybody know what $P(B_s>0,B_t>0)$ is? I think I remember it was some $arctan$-law but I don't know the exact form. So I do not need a ...
2
votes
2answers
111 views

$E[e_te_s\Delta B_t\Delta B_s]$ for $\Delta B_t$ Brownian motion increments and $e_t(\omega)$ a measurable function.

Let $\Delta B_j=B_{t_{j+1}}-B_{t_j}$ where $B_t$ is Brownian motion, and $e_i(\omega)$ measurable with respect to $\sigma(B_{t_i})$. In Oksendal's 'Stochastic Differential Equations' he states: $$ E[...
4
votes
2answers
514 views

Hitting time of an open set is not a stopping time for Brownian Motion

Let $(B_t)$ be a standard Brownian motion and $\mathcal F_t$ the associated canonical filtration. It's a standard result that the hitting time for a closed set is a stopping time for $\mathcal F_t$ ...
3
votes
1answer
192 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: $\...
2
votes
2answers
333 views

Is $(B_t^2)$ Markov where $(B_t)$ is Brownian motion?

I am pretty sure $(B_{t}^{2})$ not Markov because the squared random walk is not. Showing the square of a Markov process is or isn't Markov I guess I can repeat the method since to be Markov it ...
1
vote
1answer
171 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
0answers
111 views

Brownian motions under different filtrations, quadratic covariation, covergence of approximating sum

This question has been posted on MO for a few days and I have not received any answer. Let $(B_t^i,\mathcal F_t^i)$, $i=1,2$ be two Brownian motions. Suppose the filtration $\{\mathcal F_t^1\}$ is ...
0
votes
2answers
347 views

How to show stochastic differential equation is given by an equation

I I tried using substitution and I got an extra integral at the end and do not know how to proceed. Can anyone help me to break this down?
21
votes
3answers
4k views

Can I apply the Girsanov theorem to an Ornstein-Uhlenbeck process?

Let $W_t$ be a standard Brownian motion, and $X_t$ a measurable adapted process. Girsanov's theorem says that under certain conditions, the Brownian motion with drift $Y_t = W_t - \int_0^t X_s\,ds$ ...
17
votes
2answers
15k views

Implementing Ornstein–Uhlenbeck in Matlab

I am reading this article on Wikipedia, where three sample paths of different OU-processes are plotted. I would like to do the same to learn how this works, but I face troubles implementing it in ...
13
votes
3answers
752 views

Stochastic Integrals are confusing me; Please explain how to compute $\int W_sdW_s$ for example

I have been trying hard to understand this topic, but only failing.Reading through my lecture notes and online videos about stochastic integration but I just can't wrap my head around it. The main ...
5
votes
1answer
2k views

Solving SDE: $dX(t) = udt + \sigma X(t)dB(t)$

Solve the SDE: $dX(t) = udt + \sigma X(t)dB(t)$ Provided Question The SDE is $dX(t) = udt + \sigma X(t)dB(t)$. Find $X(t)$, where $X(t)$ is some stochastic process and $B(t)$ is a Wiener process. ...
15
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4answers
3k views

Quadratic variation of Brownian motion and almost-sure convergence

Say that $W(t)$ is a Brownian motion. The quadratic variation $[W,W](t)$ is defined in terms of a partition $\Pi = \{0 = t_0 < t_1 < \cdots < t_n = t\}$ by $$ \begin{split} [W,W](t) &= ...
6
votes
1answer
3k views

Expectation value of a product of an Ito integral and a function of a Brownian motion

this problem has come up in my research and is confusing me immensely, any light you can shed would be deeply appreciated. Let $B(t)$ denote a standard Brownian motion (Wiener process), such that the ...
9
votes
2answers
977 views

Laplace transform of integrated geometric Brownian motion

Is there any closed form of the Laplace transform of an integrated geometric Brownian motion ? A geometric Brownian motion $X=(X_t)_{t \geq 0}$ satisifies $dX_t = \sigma X_t \, dW_t$ where $W=(W_t)_{...
7
votes
0answers
475 views

proving equalities in stochastic calculus

I am struggling with this question: FIRST PART (almost done, but stuck somewhere): Let $Z $~$ N(0,1)$ be a standard normal random variable, and define a function $F$ by the formula \begin{equation} ...
3
votes
3answers
179 views

What is $\mathbb{E}[W(s)\mathrm{e}^{W(S)}]$ where W(S) is a standard Brownian Motion?

I know that since it is a Brownian motion that $W(s)\sim{N(0,s)}$. One approach I tried, was to see that $W(s)-W(0)\sim N(0,1)$ and to let $X = W(s)$ and then solve $\mathbb{E}[X\mathrm{e}^X]$ but I ...
5
votes
1answer
1k views

(Elementary) Markov property of the Brownian motion

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname{P})$, i.e. $B$ is a real-valued stochastic process with $B_0=0$ almost surely $B$ has independent ...
3
votes
1answer
1k views

Continuity of the Ito integral

Let $B_t$ be a 1-dimensional Brownian motion. I am following "Stochastic Differential Equations" by Bernt Øksendal. On the page 32 (it is displayed in the link I've put) there is a proof of existence ...
1
vote
2answers
2k views

Law of large numbers for Brownian Motion (Direct proof using L2-convergence)

In “Brownian Motion” by Schilling and Partzsc, they give a HINT to prove the Law of Large Numbers for Brownian Motion (not in their solutions, fyi) by (1) Noting that $\left|\frac{B_{t}}{t}\right|\...
1
vote
2answers
898 views

Independent increments in squared brownian motion

I'm trying to prove that the squared Brownian motion $(W_t^2)$ doesn't have independent increments, I tried using the covariance and it doesn't quite work, can anybody give me any pointer to how to ...
9
votes
1answer
2k views

Hölder Continuity of Fractional Brownian Motion

I would like to prove the following theorem: Let $H\in (0,1)$. The fractional Brownian motion $B_H$ admits a version whose sample paths are $a.s.$ Hölder continuous of order strict less than $H$. ...
7
votes
1answer
345 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 \...
4
votes
2answers
3k views

Brownian bridge [closed]

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,\...
3
votes
0answers
117 views

No drift brownian motion problem

Given two same brownian motion with no drift and different variances: $$dG_1= \sigma_1 G_1 dW $$ $$dG_2= \sigma_2 G_2 dW $$ and two barriers $P_1 > P_2$ assuming that $ \sigma_1 > \sigma_2 $ ...
2
votes
0answers
664 views

fractional Brownian motion is not a semimartingale. How to apply Ergodic theorem in the proof of this theorem?

Here is the proof of the theorem. I couldn't understand how to apply Ergodic theorem in this proof. Let $X=(X_t)_{t\geq0}$ be a fractional Brownian motion with self-similar parameter $H\in(0,1)$. We ...
1
vote
2answers
3k 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 $A_t = W_t^3 - 3tW_t$. ...
10
votes
1answer
1k views

expectation of $\int_0^t W_s^2 dW_s $ (integral of square of brownian wrt to brownian)

Looking for answers on the site I came across this answer with 3 upvotes where I am having trouble to understand an integral. Having not enough reputation (SE requires 50 reputation at least and I did ...
7
votes
2answers
2k views

Is the condition “sample paths are continuous” an appropriate part of the “characterization” of the Wiener process?

Wikipedia has separate articles on "Brownian motion" and "Wiener process" (http://en.wikipedia.org/wiki/Brownian_motion and http://en.wikipedia.org/wiki/Wiener_process ). I am not an expert, but that ...
5
votes
1answer
1k views

Joint distribution of Brownian motion and its running maximum

$B$ being standard Brownian motion, its running maximum is defined as $M_t = \sup_{0\leq s\leq t} B_s$. I am trying to follow the proof of the following result but I don't understand some of the steps ...
5
votes
1answer
981 views

To confirm the Novikov's condition

I have a question about Novikov's condition. Let $L$ be a local martingale such that either $\exp \left(\frac{1}{2}L \right)$ is a submartingale or $E[\exp\left(\frac{1}{2} \langle L,L \rangle_{t} \...