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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12
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3answers
6k 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 ...
5
votes
3answers
918 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 ...
8
votes
1answer
761 views

Calculate $\mathbb{E}(W_t^k)$ for a Brownian motion $(W_t)_{t \geq0}$ using Itô's Lemma

Show by using Ito's Lemma, for $k \geq 2$ the following result hold. $$E[W(t)^k] = \frac{1}{2} k(k-1)\int_0^t E[W(s)^{k-2}]ds$$ where $W(t) = N(0,t)$ is standard Brownian motion. I think ...
9
votes
2answers
827 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 ...
7
votes
3answers
2k 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 ...
1
vote
1answer
145 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 ...
1
vote
1answer
220 views

Brownian Motion with Optional Stopping Theorem (OST)

Let $(B_t)_{t \geq 0}$ be a standard Brownian Motion and let $T:=\inf\{t \geq 0: B_t=at-b\}$ for some positive constant $a,b>0$. Calculate $\mathbb{E}[T]$. How do i begin it?
2
votes
2answers
447 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 ...
0
votes
2answers
41 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?
13
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4answers
2k 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 ...
13
votes
3answers
3k 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 ...
3
votes
1answer
533 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 $\displaystyle H_n(t,x)=\frac{(−t)^n}{n!} e^{x^2/(2t)} \frac{d^n}{dx^n}e^{-x^2/(2t)}$. ...
12
votes
2answers
1k 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) &= ...
2
votes
2answers
756 views

First hitting time for a brownian motion with a exponential boundary

Let $B_t$ be the standard Brownian Motion. Is the distribution/density of the first hitting time of $B_t$ for an exponential decaying boundary known? Trying to be more formal, if ...
7
votes
2answers
548 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 ...
6
votes
0answers
227 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
0answers
68 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 $ ...
1
vote
0answers
40 views

Why do people all the time exploiting almost sure properties of a stochastic process as if they were sure properties?

All the time, I see people working with a given Brownian motion $(B_t)_{t\ge 0}$ on a fixed probability space $(\Omega,\mathcal A,\operatorname P)$ and suddenly exploiting its almost sure properties ...
7
votes
1answer
156 views

Must $n$ independent Wiener processes be simultaneously positive at some time?

Consider $n$ independent one-dimensional Wiener processes $(W_i)_{1\leqslant i\leqslant n}$. Is there with probability $1$ some time $t\in[0,1]$ such that $W_i(t)>0$ for every $1\leqslant ...
7
votes
1answer
761 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 $H$. ...
6
votes
2answers
953 views

Dominated convergence problems with Wald's identity for the Brownian Motion

In the course of proving Wald's second identity $E(B^2_T)=E(T)$, where $(B_t)_{t\geq0}$ is the Brownian motion and $T$ is a stopping time with $E(T)<\infty$, I got stuck with the following problem. ...
5
votes
1answer
1k 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 ...
4
votes
2answers
595 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 ...
4
votes
1answer
3k 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 ...
2
votes
1answer
98 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 ...
0
votes
0answers
63 views

Convergence in $L^2$ of the stochastic integral $\int\limits^{t}_{0}\frac{B_s}{e}1_{B_s\in(-e,e)}dB_s$

Let $e\in \mathbb{R}^+$ and $B_t$ 1-dimensional Brownian motion. Consider $$X_t=\int^{t}_{0}\frac{B_s}{e}1_{B_s\in(-e,e)}dB_s.$$ How to show that $X_t \to 0$ in $L^2$ as $e\to0$? Obviously the ...
0
votes
2answers
174 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 $A_t = W_t^3 - 3tW_t$. ...
4
votes
1answer
275 views

Applying Ergodic Theorem on fractional Brownian motion

For a fractional Brownian motion $B_H$ consider the sequence for $p>0$ $$Y_{n,p}={1\over n}\sum\limits_{i=1}^n \left|B_H(i)-B_H(i-1)\right|^p.$$ By the Ergodic Theorem it is ...
2
votes
1answer
92 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
1answer
124 views

Limit of a stochastic integral

Let $W_t$ be a one-dimensional Brownian motion and I would like to prove $$\lim_{\beta\rightarrow+\infty}\sup_{0\leq t\leq T}\left|e^{-\beta t} \int_0^te^{\beta s}\mathrm dW_s\right|=0$$ This is an ...
1
vote
1answer
61 views

Lower bound on the probability of the maximum of a reflecting Brownian motion

Let $\{W_t\}_{t\geq 0}$ be a standard Brownian motion (starting at $0$). For $T$ large enough, I would like to prove that $P(\max_{t\in[0,T]} |W_t| \leq c T^{1/3})$ is bigger than a negative power of ...
1
vote
1answer
165 views

martingale and stochastic Integral

Let ${W_t}$ be 1 dimension Brownian motion and $X_t:=\exp(t/2)\cos W_t$ $t\in[0,T]$. Show that $X_t$ is martingale. I understood $df(t,W_t)=-\exp(t/2)\sin xdW_t$ , but I don't know why it become ...
1
vote
1answer
946 views

Sum of 2 Brownian motions

Let's say, that $B_t$, $t\geq0$ is standard Brownian motion (Wiener process). Let's define process $$X_t=B_t+B_{t^2}\text{, }t\geq0$$ I need to find its variance, covariance, find out if it's ...
1
vote
0answers
85 views

What is the intuitive meaning of $K_1, K_2, K_3$ in regards to the conditional density formula derivation in Brownian motion.

In my text, there is a passage that says: "Suppose we require the conditional distribution of $X(s)$ given that $X(t) = B$, where $s < t$. The conditional density is: $$ \begin{align*} f_{s\mid ...
0
votes
1answer
86 views

Distribution of a transformed Brownian motion

Let $W$ be a standard Brownian motion. From an earlier proven result I know that $N_t = \exp\left\{a W_t - \frac12 a^2 t \right\}$ defines a martingale on the natural filtration of $W$ for all $a \in ...
0
votes
1answer
89 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 ...
15
votes
2answers
1k views

Brownian bridge expression for a Brownian motion

Let $B_t$ be a standard Brownian motion in $\mathbb R$, then the Brownian bridge on $[0,1]$ is defined as $$ Y_t = a(1-t)+bt+(1-t)\int\limits_0^t\frac{\mathrm dB_s}{1-s} $$ for $0\leq t<1$. Here ...
16
votes
2answers
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 ...
8
votes
1answer
404 views

Definition of the Brownian motion

The way I understood the definition of a Brownian motion $B_t$ in $\mathbb R$ is that it consists of two parts: We first define the finite-dimensional distributions $$ ...
12
votes
2answers
429 views

A planar Brownian motion has area zero

I'm looking for proofs of Paul Lévy's theorem that a planar Brownian motion has Lebesgue measure $0$. I know of only two proofs: one is in Lévy's original paper (Théorème 12, p. 532) and the other is ...
8
votes
2answers
678 views

Sobolev meets Wiener

Even though the Wiener process (Brownian motion) is continuous, it has no derivative at any point. Does it at least have weak derivatives?
6
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2answers
1k 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 ...
5
votes
1answer
553 views

$\int_0^tB_s^2\ dB_s$ - Gaussian Process and independent increments?

For $(B_t)_{t\ge0}$ a standard Brownian motion (Wiener process) define the stochastic process $X_t:=\int_0^tB_s^2\ dB_s$. I am currently trying to assess if $(X_t)_{t\ge0}$ is a Gaussian process and ...
6
votes
2answers
2k 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 ...
6
votes
2answers
2k views

Quadratic Variation of Brownian Motion

Quadratic Variation of a Brownian motion $B$ over the interval $[0,t]$ is defined as the limit in probability of any sequence of partitions $\Pi_n([0,t])=\{0=t^n_0<\cdots<t^n_{k(n)}=t\}$ of the ...
4
votes
1answer
124 views

Is the graph of a Brownian motion over an interval measurable?

Let $n \in \mathbb{N}_1 := \{1, 2, \dots\}$ and let $B:\Omega \times [0, \infty) \rightarrow \mathbb{R}^n$ be a standard, $n$-dimensional Brownian motion over the probability space $(\Omega, ...
1
vote
1answer
112 views

Strong solutions SDE inequality with an application of Gronwall's inequality

Suppose that we have a general SDE on a probability space $(\Omega,\mathcal{F},P)$ defined by: $$ dX_t = b(t,X_t) dt + \sigma(t,X_t) d W_t, $$ where $W$ is a Brownian motion and $b$ and $\sigma$ are ...
4
votes
2answers
362 views

Brownian motion introduction

I didn't get any answers to my previous question; so I am trying a different tack. I am familiar with a first course in probability theory using measure theory, to the extent of proving the Central ...
3
votes
1answer
236 views

How to show that $X_t = \frac{1}{\left| B_t -x\right|}\mathbb{1}_{\left\{ B_t \neq x\right\}}$ (“inverse brownian”) is a martingale?

Consider $$X_t = \frac{1}{\left| B_t -x\right|}\mathbb{1}_{\left\{ B_t \neq x\right\}}$$ where $ \left(B_{t }\right)_{t \geq 0}$ is a $ \mathcal F_t$- brownian motion in $\mathbb R ^3$, null at ...
2
votes
1answer
104 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 ...