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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Brownian motion: Show $\lim \sum W_{i} (W_{i+1}-W_{i})=\frac12 W^2_t-\frac12 t$ in probability.

Let $\{t_i\}_{i=1}^n$ be a partition of $[0,t]$ and $W$ a standard Brownian motion. Write $W_i$ for $W_{t_i}$. Show $$ \lim \sum W_{i} (W_{i+1}-W_i)=\frac12 W^2_t-\frac12 t $$ where the limit is in ...
3
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2answers
346 views

Independent increments?

The questions are simple: Does the process $ X(t) = \int_0^t B(s)ds$ have independent increments? What about $X(t) = \int_{t-r}^{t}B(s)ds$? Here $B$ denotes the standard Brownian motion. ...
3
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1answer
106 views

Show the limsup of $B_t/\sqrt{t}$ when $t\to\infty$ is positive

I am trying to prove the following statement about the standard Brownian Motion: $\varlimsup_{t\rightarrow\infty} \frac{B_t}{\sqrt{t}}>0$. I know that it is trivial to prove the above statement by ...
3
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2answers
152 views

That Brownian Motion's increments are gaussian is “not surprising”?

In section 1 of chapter 1 of Continuous Martingales and Brownian Motion, the authors claim that the fact that the increments of of Brownian motion are gaussian random variables "is not ...
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2answers
161 views

Mean and variance of this random variable

How can we compute the mean and variance of $e^{W_tW_s} $ where $(W_t)_{t \geq 0} $ is a Brownian motion? If we want to compute $ \mathbb{E}(W_tW_s) $, the usual thing to do is to assume that $ s ...
3
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1answer
2k views

Covariance of Brownian Bridge?

I am confused by this question. We all know that Brownian Bridge can also be expressed as: $$Y_t=bt+(1−t)\int_a^b \! \frac{1}{1-s} \, \mathrm{d} B_s $$ Where the Brownian motion will end at b at $t ...
3
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1answer
575 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. ...
3
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1answer
688 views

Proving the reflection principle of Brownian motion

The reflection principle of Brownian motion states that Brownian motion reflected at some stopping time $\tau$ is still a Brownian motion. The proof found in Mörters & Peres (as well as in ...
3
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1answer
56 views

two r.v sharing the same law

I have a question: Let $X=B^{+}$ or $X=|B|$ where $B$ is the standard Brownian motion. Set $$J_p=\sup_{t\geq 0}(X_t-t^{\frac{p}{2}})$$ where $p>1$ and $q$ its conjugate ...
3
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1answer
188 views

Probability for brownian motion

How can I prove it? For $b>a>0$, show that $$ \operatorname{Pr}\left({\sup_{t\geqslant 0}\left(\frac{b+X(t)}{1+t}\right)\geqslant a}\right)=e^{-2a(a-b)} $$ where $X(t)$ is a Brownian ...
3
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1answer
475 views

$\mathcal{F_t}$-martingales with Itô's formula?

I need a little help with a problem. I am given some stochastic processes and supposed to show that they are $\mathcal{F_t}-$martingales. The first one is this, and they all look similar: ...
3
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1answer
256 views

a question about Wiener process

I don't quite understand a property of the Wiener process. Such process has the property that $W(t) - W(s) \sim \mathcal{N}(0, t-s)$ where $t > s > 0$. What I don't understand is this. As the ...
3
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1answer
42 views

How to compute stochastic integral: $\int_0^t d(B_s^2)$

Here, $B_t$ is Brownian motion at time $t$ What property is used to compute the integreal $\int_0^t d(B_s^2)$? Shouldn't there be some other variable attached with the differential $d$ ?
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1answer
75 views

How to compute $E[W_t^4]$, with $W_t$ being a standard Wiener process

I want to compute the fourth moment of a standard Wiener process: $E[W_t^4]$. My solution is not equal to the one in my textbook but I don't understand where I am wrong. I started by applying Ito's ...
3
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1answer
185 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 ...
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1answer
300 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 $\displaystyle H_n(t,x)=\frac{(−t)^n}{n!} e^{x^2/(2t)} \frac{d^n}{dx^n}e^{-x^2/(2t)}$. ...
3
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1answer
86 views

Determine the distribution of $\int_0^t (W_s-\frac{s}{t}W_t) ds$, where $(W_s)_{s\geq 0}$ is a brownian motion

I have to find the distribution of $X_t:=\int_0^t (W_s-\frac{s}{t}W_t) ds$ where $(W_s)_{s\geq 0}$ is a brownian motion. I already showed the first integral $\int_0^t W_s ds$ is ...
3
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1answer
108 views

Strong Markov property of Brownian motion

I was able to understand Brownian Motion $\{B(t):t\geq0\}$ has Strong Markov Property i.e. For any stopping time $\tau$, $P(B(t+\tau)\leq y | \mathcal{F}_{\tau})=P(B(t+\tau)\leq y|B(\tau))$ a.s. , $y ...
3
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2answers
373 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 ...
3
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1answer
83 views

Given the SDE: $dX_t=dB_t+b(X_t) dt$ with $(x,b(x)) \leq 0, \forall x \in \mathbb{R}^n$, prove that $E[|X_t|^2] \leq nt+E[|X_0|^2]$

I'm working on this problem: Given a solution $X_t$ to the SDE $$dX_t=dB_t+b(X_t) dt$$ where $B_t$ is an $n$-dimensional Brownian motion, and $b:\mathbb{R}^n \to \mathbb{R}^n$ a Lipschitz ...
3
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1answer
125 views

Are these 2 random variable independent???

Assume $\{B_t:t\ge0\}$ be a brownian motion process. Is $B_s-\frac{s}{t}B_t$ and $B_t$ independent given that ($s\le t$)
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1answer
59 views

$E \left\{ \left( \sum_{i=0}^{n-1} \left[ B_{c_i} \left( B_{t_{i+1}} - B_{t_i}\right)\right] \right)^2 \right\}$, where $c_i \in [t_i, t_{i+1}]$

Let $B$ be a standard Brownian motion and $\{t_i\}_{i=0}^n$ a partition of $[0,t]$. Define $c_i= (1-c)t_{i+1}+ct_i$, for some $c \in [0,1]$. Write $B_i$ for $B_{t_i}$ and $$ S_n=\sum_{i=0}^{n-1} ...
3
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1answer
1k views

To show that a given process is Gaussian

Suppose I have given a Brownian Motion $W$, this is a Gaussian process, and I define: $$B_s:=W_{t-s}-W_t$$ for $0\le s\le t$. Clearly this random variable has expectation zero. For the covariance ...
3
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1answer
272 views

Are hitting times of Brownian motion independent?

Suppose that $B_t$ is a standard Brownian motion. And $T_a$, $T_b$ are the hitting time whereas $a<0$, $b>0$. Then are these two random variables independent?
3
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1answer
46 views

How to calculate $\mathbb{E}((B_3-B_2)(B_4-B_{\pi}) \mid B_1)$ for a Brownian motion $(B_t)_{t \geq 0}$

I want to find the best predictor of $(B_3-B_2)(B_4-B_{\pi})$ given an observation of $B_1$ Where $B_t$ is brownian motion for time $t \geq 0$. I am not sure how to approach this. I know it will be ...
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1answer
44 views

What does $s$ and $t$ stand for in this definition of fractional brownian motion?

$$B_H(t_2,\omega)-B_H(t_1,\omega) = \frac{1}{\Gamma(H+1/2)}\Bigg\{\int_{-\infty}^{t_2}(t-s)^{H-1/2}dB(s,\omega)-\int_{-\infty}^{t_1}(t-s)^{H-1/2}dB(s,\omega)\Bigg\}$$ It's taken from Mandelbrot & ...
3
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1answer
81 views

What is the explicit obstruction to almost sure convergence in stochastic integrals?

Let $B(\omega,t)$ be a Brownian motion defined on some appropriately filtered probability space $(\Omega,\mathcal{F}_{t},\mathbb{P})$, and let $f(\omega,t)$ be a stochastic process defined on $\Omega$ ...
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1answer
56 views

Determining if some random variable is a stopping time

I am stuck on this issue: Let $(B_t)$ be a Brownian motion. We know that since $\{0\}$ is a closed set in $\mathbb{R}$ and that $(B_t)$ is a continuous adapted process, $$ \tau:= \inf \{ t\geq 0 : ...
3
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1answer
26 views

Independence of random variables involving Brownian motion

I am reading a book on stochastic analysis and I don't understand the following (i.e. don't know how to prove it rigorously): Let $B$ be a standard Brownian motion and $\{ \mathcal{F}_t \}$ be the ...
3
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1answer
106 views

perfectly correlated processes

I am really stuck in this question: Let $\{S_t\}$ and $\{S'_t\}$ be two stochastic processes, satisfying \begin{equation} dS_t = S_t ( \sigma_t \,dB_t + r_t \,dt), \quad dS'_t = S'_t (\sigma'_t ...
3
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1answer
38 views

Series of independent gaussian variables and brownian motion

I am checking the proof of the construction of a brownian motion in $[0,\pi]$. We show that \begin{gather*} t \mapsto B^m_t = \frac{t}{\sqrt{\pi}}X_0 + \sqrt{\frac{2}{\pi}}\sum_{n=1}^{2^m-1}X_n ...
3
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1answer
45 views

Independence of increments of a pair of independent Brownian motions

Suppose I have two Brownian motions $X$ and $Y$, which are independent. In other words, for any finite set of times $0 < t_1 < t_2 < \cdots < t_n$ the random vectors $(X(t_1),\ldots , ...
3
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1answer
75 views

Ito formula applied to $\frac{1}{t}\int_0^t W_s ds $

I got this expression and I have to calculate its differential by the Ito formula, $W_t$ denotes the Brownian motion: $$\frac{1}{t}\int_0^t W_s ds $$ I calculate the derivative of ...
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1answer
121 views

Integrating brownian motion times exponential function

I am trying to calculate $$\int_0^tB_se^{\lambda s}ds$$ but I am unsure of how to start the computation. The motivation behind this is that I read (and am now trying to prove) that ...
3
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1answer
240 views

Expectation of brownian motion at hitting time

Am i correct in my derivation? I want to calculate $\mathbb{E}B_{\tau_a}$. From the definition of the hitting time i get $B_{\tau_a}=a$, so $$\mathbb{E}B_{\tau_a}=\mathbb{E}a=a$$ I am new to the ...
3
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1answer
92 views

The limit supremum of a function involving Brownian motion

I would like, for some $\delta>0$ and a Brownian motion $B$, to calculate $\displaystyle\limsup_{t\to\infty}\left(\exp\left( (1+\delta)t\right)\cdot\exp\left(-B_t-\frac{t}{2}\right)\right)$ ...
3
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2answers
1k views

What is the definition of a sample path of Brownian motion?

My question has been asked before at beginner's question about Brownian motion . There was only one answer, which was not accepted. It was probably incorrect, because nothing was said about ...
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1answer
86 views

Distribution of a Brownian motion with respect to $\mathbb{P}^x$

Let $(\Omega,\mathcal{A},\mathbb{P})$ a probability space and $(B_t)_{t \geq 0}$ a Brownian motion (started in $x=0$). Then one can define a probability measure $\mathbb{P}^x$, $x \in \mathbb{R}$, on ...
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1answer
136 views

Brownian motion interesting question

I found this interesting question on the internet, but unfortunately I could not solve it. What is probability that Brownian motion (starting at origin) has value 1 before having value -2?
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1answer
151 views

Autocorrelation of wrapped Wiener process

Let $\phi(t)$ be a Brownian Walk (Wiener Process), where $\phi\in[0,2\pi)$. As such we work with the variable $z(t)=e^{i\phi(t)}$. I would like to calculate $$E(z(t)z(t+\tau)).$$ This is equal to ...
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1answer
434 views

Brownian Motion and Probability Density Function

I'm confused with this particular problem. B(t) is a BM. MT is the maximum of B(t) in [0,T]. What is PDF?
3
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1answer
572 views

Proving that a process is a Brownian motion

Let $B$ be a Brownian motion with natural filtration $(\mathcal{F}_t)_{t\geq 0}$ and let $\mathcal{H}_t$ be the $\sigma$-algebra generated by $\mathcal{F}_t$ and $B_1$. Define $$A_t = ...
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1answer
34 views

Distribution of Brownian Motion help

If $X = \frac{B_1 - B_3 + B_2}{\sqrt{2}}$ Where $B_t$ is brownian motion at time $t$. And I want to find the the distribution of $X$, how would I do so? $E[X] = 0$ is fairly straight forward. For ...
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1answer
66 views

Conditional expectation of Wiener process

I want to calculate $E(W_t | W_1)$, $E(W_t^2 | W_1)$ and $E(W_t^2 | W_1, W_2)$, where $(W_t)_{t\geq0}$ is a Wiener process. For the first one I used the conditional distribution formula for the ...
3
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1answer
53 views

Hitting time process of Brownian motion [closed]

I am stuck with this problem: Let $(B_t)$ be a standard Brownian motion in $\mathbb{R}$. For $t \geq 0$, let $$ H_t = \inf \{ s \geq 0 : B_s = t \}, \quad S_t = \inf \{ s \geq 0 : B_s > t \}. $$ ...
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): ...
3
votes
1answer
110 views

Expectation of Square of Stopping Time

Let $B_t$ be standard Brownian motion and $a < 0 < b$. Define stopping time $T$ as follows. $$T = \min \{t \geq 0: B_t \in \{a, b\} \}.$$ The expectation of $T$ is $\mathbb ET = |a|b$ and can be ...
3
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1answer
75 views

Extended (or augmented) stopping times

I am trying to prove that $\tau$, defined as: $$ \tau = inf\{t > 0 \mbox{ }|\mbox{ } B_t < t-1 \} $$ is a stopping time with respect to the filtration $(\mathscr{F}_{t+}^B)_{t\geq 0}$ where ...
3
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1answer
64 views

Convergence in $L^2$ and proof of Brownian motion

Could anybody give me some hints on the following question? I was doing some exercises on Brownian motion and found this online: Let $\left \{ X_n \right \}_{n=1}^\infty$ be a sequence of ...
3
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1answer
617 views

Expectation of stochastic integrals related to Brownian Motion

I'm trying to solve a problem that's now doing my head in a bit. I'll share with you the question and let's see if somebody can shed some light into the matter: Let B be a standard Brownian Motion ...