1
vote
0answers
13 views

Showing $E[e^{-\lambda \tau_{a}\wedge\tau_{-a}}]=sech(a\sqrt{2\lambda})$

This is homework so no answers please. For $\tau_{a}=inf_{t}(B_{t}=a)$ , we already know $E[e^{-\lambda \tau_{a}}]=e^{-\sqrt{2\lambda}a}$. By $B_{t}$ I mean Brownian motion. The question is to show: ...
-1
votes
0answers
16 views

weiner process expected max given current state? [on hold]

Suppose we have weiner process over interval t=0 to t=1, with std. 1 over interval and mean 0. If time now is r where r on interval 0 to 1 and x is current value of process and m is current maximum ...
1
vote
1answer
21 views

mean hitting time of a level and growth rate of maximum process

Let $X_t$ be the absolute value of Brownian motion starting at $0$, let $\tau_x$ be it's first hitting time of the level $x>0$, and let $M_t$ be it's running maximum up to time $t$. Suppose we knew ...
1
vote
1answer
21 views

Probability Brownian Motion doesn't hit a point in the limit.

This is a question from Revuz and Yor (exercise 3.18) for which I seem to get a different answer. Show that $\lim_{t \to \infty}\,t^{1/2}\,\mathbb{P}\{B_s\leq1\,\forall\, s\in[0,t]\}=\sqrt{2/\pi}$. ...
1
vote
0answers
10 views

Holder continuity, brwonian motion [duplicate]

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 ...
0
votes
0answers
22 views

Brownian motion and proximity to a set

Here is the problem: Given two square planes $S_{1},S_{2}\subset \mathbb{R}^{3}$ oppositely positioned to the origin along the x-axis and $S_{1},S_{2}\perp x$-axis. Also, let ...
-1
votes
1answer
36 views

Variance of the increment in Brownian Motion

How do you show that variance of $B_t-B_s$ is $ t-s$. where $B_t$ = $\int p(t,x,y)dy$ where $p(t,x,y)$ = $ (1/ \surd(2\pi t))\exp(-(x-y)^2/2t$ where t>0 and $x,y \in \mathbb{R}$
2
votes
1answer
40 views

Invariance Properties of Brownian Motion

I am trying to make sense of the Scaling-Invariance and Time-Inversion properties of Brownian motion by producing a sample path. For the record, I am using the following definitions. Let $B(t)$ be the ...
4
votes
2answers
147 views

A counter example of Brownian Motion

Here is an example in my textbook to illustrate why we need the continuous sample path in the definition of Brownian motion. Let $(B_t)$ be a Brownian motion and $U$ be a uniform random variable on ...
2
votes
1answer
24 views

Filtration from a Brownian Motion

The textbook I am reading defines the filtration induced from a Brownian Motion as follows. Let $\{B(t): t \geq 0\}$ be a Brownian Motion defined on some probability space, then we can define a ...
1
vote
0answers
14 views

Stopped supremum of the Brownian local time still $L^p$ bounded in space?

Let $B_t$ be a standard Brownian motion and $L_t^x$ its local time in $x$ at time $t$. For fixed $t$ and $p>1$, it holds that $$ \sup_{x \in \mathbb{R}} \operatorname{E} [ (L_t^x)^p ] < ...
0
votes
1answer
25 views

(Multidimensional) Standard Brownian Motion: Convergence

Relating to this question, I have a further one, and hope, someone can help me. I know that $$\left(X_j - X_{j-1}\right)_{j=1}^t \xrightarrow{d} \left(Y_j\right)_{j=1}^t.$$ Further, we know that ...
0
votes
1answer
23 views

convergence to standard brownian motion

Could you help me with the following: I have that $$T(x):=\frac{X(nx)-E[X(nx)]}{\sqrt{n}} \xrightarrow{d} N(0, \frac{x^k}{k})$$ for each fixed $x>0$, where we also have that $\frac{X(nx)}{t}$ is ...
3
votes
0answers
54 views

Brownian Motion and stochastic integration on the complete real line

I'm struggling to understand stochastic integration over intervals containing zero, i.e. integrals of the form $\int_{a}^{b} X_s \, d B_s$ where $-\infty \leq a < b \leq \infty$, $(X_t)_{t \in ...
0
votes
1answer
31 views

Proving that a process has the Markov property

Let $X_t=xe^{ct+aB_t}$ where $B_t$ is one dimensional Brownian motion. How would I prove this is a Markov process using the expectation definition of a Markov process, i.e., ...
1
vote
1answer
47 views

Property of Wiener process sample path

What is a mean of time, when the trajectory of wiener process $W_t$ is over the line $y=t$? We need to find $\mathbb{E}\tau$, where $\tau=\sum\limits_{a,b:\forall t\in(a,b) ; ...
4
votes
2answers
99 views

Distribution related to brownian bridge

Let $B(t)$ be a Brownian Bridge and $U$ is uniformly distributed on $(0,1)$. I wish to know the distribution function $B(U)$. Is it possible? As we know, $B(t)\sim N(0,t(1-t))$. But, I haven't a clue ...
0
votes
0answers
117 views

Property of G- Stochastic Calculus

i have maybe a stupid question about an equation. It is said that \begin{equation} \inf\limits_{P \in \mathcal{P}}\mathbb{E}_{P} \left[\int_0^T \varphi_{x}(t,X_{t})X_{t}\pi^{T}_{t}\,\mathrm ...
0
votes
0answers
37 views

Brownian Motion first hitting time distribution

I have a question concerning the distribution of the first hitting time of Brownian Motion $\tau_x = \inf_{t\geq 0}\{W_t=x\}$, where $W_t$ is Brownian motion. Using some calculus, I found out that the ...
0
votes
0answers
14 views

Modulus of continuity of maximum of Brownian motion

Let $B(t)$ be the standard Brownian motion and $M(t)$ its maximum process, i.e. $M(t) = \sup_{0\leq s\leq t}B(t)$. What can be said about the modulus of continuity of $M(t)$?
1
vote
1answer
56 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 ...
0
votes
1answer
46 views

Distribution of Sum of Brownian Motion and Integrated BM

Let $W(t)$ be a standard Brownian motion (BM), in particular $W(t) \sim \mathcal{N}(0,t)$. Then it is easily shown that $\int_0^T W(t) dt \sim \mathcal{N}(0, T^3/3)$. Question: What is the ...
1
vote
0answers
55 views

Variance of Integrated Geometric Brownian Motion

I'm just asking for verification that my derivation is correct, as I can't seem to find this result elsewhere. I'd like to calculate $Var(\int_0^T X(t) dt)$ where $X(t) = X_0e^{(\mu - ...
0
votes
0answers
30 views

Expectation Involving Two Values of Geometric Brownian Motion

Not sure this is the best place to ask for verification, but I can't seem to find a derivation anywhere else. I want to calculate $\mathbb{E}[e^{\sigma(W_t + W_s)}]$, where $W_t$ and $W_s$ are two ...
1
vote
1answer
37 views

Property of Brownian Motion's paths

We are considering a Brownian Motion $(B_t)_t$ with values in $\mathbb{R} $ starting from $x$ defined on the stochastic basis: $$(\Omega,\mathcal{E},(\mathcal{F}_t)_t,\mathbb{P}^x)$$ Then, let's ...
6
votes
0answers
184 views

Does this random variable have a density?

I have a persistent problem, which I'm almost certain can be answered using elementary probabilistic arguments, but for some reason I've been stuck for some time. Here is the problem. Let $(B_s, s ...
1
vote
1answer
40 views

Combination of Wiener Processes

If $W_s$ and $W_t$ are wiener processes, we have that the probability that $W_s$ and $W_t$ attain maximum is (I am concluding this from "running maximum", but I am not sure) ...
0
votes
1answer
26 views

Probabilistic solution Poisson problem

Let us consider the Poisson problem \begin{cases} \frac{1}{2}u''=-f\qquad\text{in}\,\,(a,b)\\u(a)=u(b)=0 \end{cases} where $f:(a,b)\to\mathbb{R}$ is continuous and bounded. We have obtained ...
0
votes
0answers
29 views

Strong law of large numbers for a Brownian motion

I have a question in regard to the strong law of large numbers. It is well known that, if $B_t$ is a Brownian motion, then $\displaystyle\lim_{t\to\infty}\frac{B_t}{t}=0$. This had me wondering... ...
1
vote
1answer
33 views

Reflection Principle interpretation

Given a standard Brownian motion $(\Omega,\mathcal{F},(\mathcal{F}_t)_t,\mathbb{P},(B_t)_t)$ (the standard filtration $(\mathcal{F}_t)_t$), we define $$\forall t\ge 0: M_t:=\max_{0\le s\le t} B_s$$ ...
0
votes
1answer
79 views

Sample continuity of Brownian motion

I wanted to know if the Brownian motion and the fractional Brownian motion are almost surely sample continuous or not? Many thanks.
0
votes
0answers
21 views

Assume b$\gt$0,how to show sup($B_t-bt$) is a exponential random variable with parameter 2b?

Assume b$\gt$0,how to show sup($B_t-bt$) is a exponential random variable with parameter 2b? The drift here is a big touble for me.
1
vote
1answer
60 views

$B_t$ is a standard Brownian motion, show $Y=\int_0^1f(s)B_sds$ is normal and what is $var(Y)?$

$B_t$ is a standard Brownian motion, $f(t)$ is a continuous function on $[0,1]$. $Y=\int_0^1f(s)B_sds$. How to show $Y$ is normal. And what is the variance? I know I can use characteristic function ...
1
vote
1answer
36 views

Show that Px$(B(s)\ge0 $ for all 0 $\le s \le t$ and B(t) $\in$ M) = Px(B(t) \in M)$-$P-x$(B(t) \in M)$,where B(s) is brownian motion,x>0.

I want to show Px$(B(s)\ge0 $ for all 0 $\le s \le t$ and B(t) $\in$ M) = Px(B(t) \in M)$-$P-x$(B(t) \in M)$ where,x>0,M is measurable set in [0,$\infty$). The difficulty for me is how to handle ...
1
vote
0answers
73 views

Donsker for randomly stopped processes

A question regarding Donsker's invariance principle. Donsker states that if $X_1, X_2, ...$ are independent and identically distributed with mean $0$ and variance $\sigma^2$ and if $S_t^n$ is the ...
0
votes
2answers
59 views

Find the generator of Markov Process

Homework question: Consider the Markov process $X_t=B_t-t^2+t$ where $B_t$ is the Brownian motion. Find the generator $Q$ of this process. I am completely confused how to find the generator for ...
3
votes
1answer
106 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 ...
0
votes
0answers
79 views

Probability calculations with integral of Geometric Brownian Motion

I am looking for the probability $Prob[A_t<A_0]$, where $A_t$$=$$A_0$$+$$\int_0^t$$S_0$$e^{(\mu-(1/2)\sigma^2)s+\sigma W_s}$$ds$$-$$c$$t$ The expression $S_0$$e^{(\mu-(1/2)\sigma^2)t+\sigma ...
1
vote
2answers
90 views

Does a Brownian motion remain in any given open set for a given interval of time with positive probability?

Let $B$ be a standard $d$-dimensional Brownian motion. Given $b>a>0$ and an open ball $U$ in $\mathbb{R}^d$, I want to be able to comment on the probability that $B$ remains in $U$ during the ...
2
votes
0answers
60 views

Why is a brownian motion conditioned to stay positive a Bessel-3

I am told this result long ago but I still don't know how to prove it. Is it because that this conditioning can be turned into a Girsanov probability change? Or is there any simpler ways to see it?
0
votes
0answers
25 views

How to understand this equation for brownian motion

I am reading this article from the notes 'an intro to SDE'. Here I dont know why in (1) he take that integral from - infinity to infinity. I mean why we do that? I just dont know what the physics or ...
0
votes
1answer
38 views

Expectation of a product of (many) 1-dimensional Brownian motions.

Let $0=t_0<t_1<t_2<\ldots$ be a sequence of positive reals. Denote by $B(t)$ the 1-dimensional Brownian motion with time $t$. It is easy to show the the expectation of the product of two ...
2
votes
1answer
66 views

Expectation of Integrals of Brownian Motion

Hello I am not a native english speaker so please let me know if something does not make sense. I am interested in computing the following: $$E\int_0^T(B_s(\omega,t))^4dt$$ Or at least showing it is ...
1
vote
2answers
59 views

$4^{Brownian(t)}$ martingale proof

Let $B(t)$ a Brownian motion. I like to prove that $4^{B(t)}$ = martingale I rewrote the expression into an exponential form (like $\exp(\ln(4) B)$), but then I don't know how to proceed.
1
vote
2answers
66 views

$\mathbb{E}[B^4(t)]$ with $B$= brownian motion

Can anyone help me to find: $\mathbb{E}[B^4(t)]$ where $B$ is a brownian motion? I thought using this density function: $f_{B_t}(x) = \frac{1}{\sqrt{2 \pi t}} e^{-\frac{x^2}{2t}}$, but I don't know ...
3
votes
1answer
86 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 ...
0
votes
2answers
56 views

two brownian motions in $ \mathbb{Z}^2 $

I was wondering what is the probability for 2 brownian walkers coming from 2 different initial positions to be at the same position at time t. I consider that at each step, each point can ...
1
vote
0answers
57 views

Probability that the value at time T from one geometric Brownian motion process is greater than the value from another GBM

I am having a competition between $n$ people (starts at time $t$=0), each who accumulates points on a daily basis, which I assume is a geometric Brownian motion process with parameters $\mu_i$, ...
2
votes
1answer
212 views

Brownian Bridge as a Gaussian Process

Let $B=\{B_t:t\geq 0\}$ be a standard Brownian motion. Define the Brownian brige $X=\{X_t:t\geq0\}$ as $$ X_t=B_t-tB_1\quad t\in[0,1] $$ Show that $X$ is (i) Gaussian and find its (ii) mean and (iii) ...
0
votes
0answers
63 views

Indicator function of an infinitesimal set

While reading a paper related to functional of brownian motion I came across the following notation $1(B_t \in dx)$, where $1(A)$ is the indicator function of the set A, and $B_t$ is a standard ...