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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1answer
143 views

Brownian motion, reproducing kernel Hilbert space, and the Laplace operator

Consider the standard Brownian motion on $[0,1]$: $$ dB_t, \; B_0 = 0, $$ defined on the probability space $(\Omega, P)$. It covariance function is $K(s,t) = \min \{s , t\}$ on $[0,1] \times ...
0
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1answer
36 views

Markov property of Brownian motion

There are two statements about Markov property: $B_t $ is Brownian motion and $\mathcal{F}$ is generated by $B$ If $s>0$ and $Y$ is bounded and measuable, then ...
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0answers
83 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 ...
0
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0answers
28 views

If a stochastic process follows Geometric Brownian Motion, does it imply that it is Log-normally distributed and vice-versa?

This might be a naive question, but it doesn't stop haunting me. Wiki page for GBM writes the SDE for GBM process and shows it follows log-normal distribution. Is it true every time or are there any ...
0
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1answer
37 views

brownian motion scaling

I have the following probability : $P( W(t) > 0 \mbox{ and }W(2t) > 0)$ on some textbook it is claimed that this is equal to $P( W(1) > 0 \mbox{ and }W(2) > 0)$ due to the scaling ...
0
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2answers
142 views

quadratic variations of Brownian motion squared

I'm trying to refresh my memories about stochastic processes. We know that Brownian motion has as quadratic variation equals to t. What is the quadratic variation of the Brownian motion squared ? ...
1
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1answer
22 views

Prove that $B \in \Lambda_\text{loc}^2 $ if $B=(B_t)_{t \in \mathbb{R_+}}$ is a real valued B.M

I know that $\Lambda_\text{loc}^2=\{\phi $ is progressive $: \forall t \geq 0,\int_0^t \phi_s^2 \, ds < \infty\text{ a.s.} \}$ Since B.m $B_t$ is almost surely continuous and ...
3
votes
1answer
59 views

Inequality for the expected values of norm of stochastic processes

Let $\underline{X}=(x_1, x_2, x_3), \; x_i \sim \mathcal{N(0,1)}$ i.i.d. For any fixed $t>0$ and $\underline{X}_0$ prove that the following holds ($\Vert\cdot\Vert$ is the Euclidean norm): ...
1
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0answers
50 views

Optional Sampling Theorem Application

Let x, y > 0. Define the first passage time of a Brownian motion $W_t$ as $\tau_a$ = min{t $\ge$ 0: $W_t$ = a}. I need to show that $$E[e^{-u\tau_x}1_{\tau_x < \tau_{-y}}] = ...
0
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1answer
31 views

A Estimation about Hölder condition

Let $p:[0,\inf) \to \mathbb{R}$ be a contionous function such that $p(0)=0$ Fix $a>1/2 , k$ is a positive integer $>\frac{1}{a-\frac{1}{2}}$. Suppose for all $n \in \mathbb{N}$ and $\lambda ...
2
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1answer
30 views

distribution of $\sup\limits_{0\le t \le 1}|W(t)|$

My prof on class told us that distribution of $S=\sup\limits_{0\le t \le 1}|W(t)|$ has been well studied, where $W$ is a Wiener process, but I need a table to find $c$ such that $P(S < c) = 0.95$. ...
2
votes
1answer
30 views

Showing that if $B_t$ is a Brownian motion then $t B_{1/t}$ is Gaussian

I want to show that if $B_t$ is a Brownian motion then $t B_{1/t}$ is a Gaussian process, i.e. that it has increments which have the normal distribution. It seems like a trivial fact, since the ...
1
vote
1answer
42 views

Probability of Wiener process hitting a particular point at an independent stopping time

Assume we have a stopping time $T$ that is independent of a Wiener process $W$. If $T$ were taking discrete values (let's say in $\mathbb{N}_0$), one can easily show (using the independence and the ...
1
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0answers
27 views

Level sets of a Wiener process

Assume we have a Wiener process $W$ starting at $W_0=0$. What can one tell about the Lebesgue measure of "level sets" $A_y = \{t>0; W_t=y\}, y \in \mathbb{R}$? I actually need to estimate these ...
0
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1answer
61 views

Hausdorff Dimension for Brownian motion over [0,1]

I am trying to calculate Hausdorff dimension for the trajectory of Brownian motion over $[0,1]$. I read the book of Morters and Peres and know that the dimension will be $\frac{3}{2}$. I tried to use ...
2
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0answers
37 views

How to calculate probability of an event in a stochastic setting?

Let $\left(\, B_{t}\,\right)_{t\ \geq\ 0}$ be a Brownian motion. Calculate the probability of the event: $$ E\equiv\left\{\,\exists\ \epsilon > 0 : \forall\ 0 < h < \epsilon, \max_{t\ \in\ ...
0
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1answer
63 views

Applying Ito's formula

This is probably an easy question but I am getting aquanted with Ito's formula and stuck on an exercise in my textbook. Let $X_{t}=W_{t}-a t/2$ where $a$ is a real number and $W_{t}$ is brownian ...
0
votes
1answer
48 views

Derivatives pricing w/ squared and cubed stock prices

I have an assignment in which $S_t$ is a stock price following a geometric Brownian motion. The task is now to show that at time t the risk-neutral price of a derivative on $S_t$ that pays $S_T^3$ at ...
1
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0answers
140 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 ...
1
vote
0answers
61 views

What is the distribution of the area between a Brownian Bridge and the x-axis?

Lets say that we have a Standard Brownian Bridge ($\sigma=1$) with endpoints $(0,0),(1,0)$ Is there a way to derive the distribution of the area between a sample path of this bridge and the x-axis?? ...
2
votes
1answer
77 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 ...
2
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0answers
61 views

Integral of a geometric Brownian motion [duplicate]

I would like to compute $G$ defined as follows $$G(t):= \exp(-\int _0^t h_s~ ds )$$ with $h$ being a geometric Brownian Motion. For that I would need first to compute $$\int_0^t ...
3
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0answers
33 views

Brownian motion, modifications vs indistinguishablity

In Protters 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 ...
1
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1answer
31 views

compactness criterion for random variables in L2

Suppose $X_n$ is a sequence of random variables such that their second moments are uniformly bounded. I would like to know a compactness criterion for this case. In analysis, if $K$ is a bounded ...
-2
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1answer
39 views

Differential of two geometric brownian motions

I am currently taking a finance course which includes some math that is currently above my level, it is however not a pure math class and we are just supposed to be able to apply the math to the given ...
0
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1answer
28 views

Harmonicity of the expectation of a stopped Brownian Motion

Let $\mathbb{E}_x$ be the expectation associated with a probability measure such that $B_{t\geq0}$ is a Brownian motion started in x. I want to show that for $D\subset\mathbb{R}^2$ bounded, $y\in D, ...
5
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0answers
49 views

Accelerated Eigenfunction Expansions of Random Functions

I am interested in eigenfunction expansions of random functions. We know that the autocorrelation of brownian motion, $\{ B_t \}_{t \geq 0}$, is given by $$ E[B_t B_s] = \min\{s,t \}, $$ which can ...
1
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1answer
34 views

Resource on Pathwise Computations Involving Brownian Motion

Let $B_{t}(\omega)$ be a standard Brownian motion on $(\Omega,\mathcal{F},\mathbb{P})$. I read in a footnote recently that almost surely the quadratic variation ...
6
votes
1answer
104 views

Converge to Brownian Motion problem

Consider the following sequence of SDEs: $dX^n_t = \sin(nX^n_t)dt + dW_t, X^n_0 = 0\,\,\,$ Show that the solutions $X^n$ converge in finite dimensional distribution to Brownian Motion. I have been ...
4
votes
1answer
457 views

Sum of Brownian Motions

I've got a little problem: if $X_{t}$ and $Y_{t}$ are two indipendent Brownian motions, is then $$Z_{t}:=X_{t}+Y_{t}$$ a Brownian motion too? I've got some troubles only with showing that $Z_t$ is ...
0
votes
0answers
21 views

Random starting point for Brownian motion

The hitting probability for balls centered at origin is $P_{x}(T_{B_{r}(0)}<\infty)=\frac{r^{d-2}}{|x|^{d-2}}>0$ where $|x|>r$. Now consider hitting time $T_{A}$ of sphere A disjoint from ...
1
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2answers
27 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: $$ ...
0
votes
1answer
50 views

Expected Return, Expected Value, and an Ito Process

I am reading John Hull's "Options, Futures, and Other Derivatives". I am currently in Ch. 31 on the HJM Model. Hull makes a statement which a need an explanation for. First, some notation. Let ...
0
votes
1answer
55 views

Transition density of Brownian bridge using generators

Let $X_{t}:=(1-t)\int_{0}^{t}\frac{1}{1-s}dB_{s}$. This satisfies SDE: $$dX_{t}=-\frac{X_{t}}{(1-t)}+dB_{t}$$ So the generator will be $A(f)=\frac{-x}{1-t}f'+\frac{1}{2}f''$ and so I think the pde ...
0
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0answers
24 views

Estimate on the Positive probability of not hitting finite measure sets in $\mathbb{R}^{d}$

In $d\geq 3$, we have that BM is transient a.s. i.e. $\lim_{t\to \infty}|B_t|=\infty$. But does this imply $1-P_x(T_A<\infty)>0$ for Borel sets $A\subset \mathbb{R}^d$ with ...
0
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1answer
34 views

A Question on the Scaling Invariance of Brownian Motion

I read the following paragraph. Let $B_t, \ t \in [0, \infty)$ be a standard linear Brownian motion. For each $q > 4$, define the following sequence of sets. $$ \Omega_k := \left\{\omega \in ...
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0answers
54 views

Cameron Martin Theorem

I am struggling with two versions of the Cameron Martin Theorem. 1) We define the measure spaces $(\Omega,\mathcal{F},P)$ and $(C[0,1],\mathcal{C},\mathbb{L}_0)$, where $\mathcal{C}:=\sigma(f\mapsto ...
1
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1answer
32 views

Zero hitting probability for positive measure sets in $\mathbb{R}^{d}$

In $d\geq 3$, we have that BM is transient a.s. i.e. $lim_{t\to \infty}|B_{t}|=\infty$. But does this imply $P_{x}(T_{A}<\infty)=0$ for some type of Borel sets $A\subset \mathbb{R}^{d}$ with ...
3
votes
1answer
301 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 $\displaystyle H_n(t,x)=\frac{(−t)^n}{n!} e^{x^2/(2t)} \frac{d^n}{dx^n}e^{-x^2/(2t)}$. ...
0
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1answer
33 views

Equivalent Stopping Times for Brownian Motions

For standard Brownian motion $B$, define stopping time $T_1:=\inf\{t>0: B_t = 3\}$ and $T_2:=\inf\{t>0: B_t = -3\}$ and $T_3 := \min\{T_1, T_2\}$. Can I say that $T_3 = \inf\{t>0, B_t \in ...
0
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1answer
27 views

Testing numerical solvers with analytic solution to Ornstein-Uhlenbeck SDE?

I have an SDE I want to solve numerically that is fairly close to the Ornstein-Uhlenbeck process: $$ dx_t=θ(μ−x_t)dt+σdW_t $$ which has analytic solution $$ ...
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0answers
29 views

Probability that Brownian motion hits both hemispheres

The problem is to find: $P_{x}(\{T_{B_{1}}<\infty\}\cap \{T_{B_{2}}<\infty\})$ where $B_{1},B_{2}$ are the two hemispheres of sphere S shown below. There are two possible paths for first ...
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1answer
115 views

Brownian Motion inequality (related to Dvoretzky-Erdoes test)

i have the following question: Let $B(t)$ be a d-dimeansional Brownian motion $d\ge 3$, and $f$ be a monoton increasing function from the positive reals to the positive reals. Let $A_n=(\exists t\in ...
0
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1answer
236 views

Is this Stochastic integral a martingale ?

Let $(B_t)$ be a Brownian motion and set $X_t = \int_0^t B_t^2 dB_s$. Is $X_t$ martingale? My idea is to rewrite $X_t$ in terms of Ito's Formula $(f(x) = \frac{1}{3}x^3)$ $X_t = \int_0^t B_t^2 dB_s ...
0
votes
1answer
24 views

Brownian Motion and Progressive Process

Let $B_t$ be a Brownian motion. Define sign function as follows. $sign(0) = 0$ and $sign(x) = \frac{x}{|x|}, \forall x \neq 0$. I do not know how to show the following two questions, especially on the ...
0
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1answer
44 views

Stopping Time and Brownian Motion [closed]

Let $B_t$ be a Brownian motion. Let $a < 0 < b$. Consider $\tau: = \min\{T_a, T_b\}$ where $T_a := \inf\{s \geq 0: B_s \leq a\}$ and $T_b := \inf\{s \geq 0: B_s \geq b\}$, namely, the first ...
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vote
1answer
81 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 ...
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0answers
10 views

Showing that $P(\sigma<\infty)=e^{-2am}$, where $\sigma=\inf_{t>0}\{B_{t}=mt+a\}$

This is homework so no answers please Any mistakes: We showed that $X_{t}=e^{\lambda B_{t}-\frac{\lambda^{2}}{2}t}$ is a martingale for any $\lambda>0$. So let $\lambda=2m$, then ...
1
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1answer
74 views

Variance process of stochastic integral and brownian motion

Let $(W_t)$ be a Brownian motion with respect to a filtration $(\mathcal{F}_t)$. For all $t \geq 0 $ set $$X_t = \int_0^t W_s^2 \mathrm{d} W_s,\qquad Y_t = W_t^7.$$ Find the covariance process ...
0
votes
2answers
40 views

Brownian and Brackets

A continuous martingale with deterministic bracket must be a Brownian motion. Is this statement ture or not, please? If true, how to show it? If not, what is a counter example?