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
627 views

Min of two stopping times is also a stopping time.

Preface: I'm having trouble with the correct solution. The Original Question: Given that $\mathscr{F}_t$ is a filtration that satisfies all the usual conditions, and given ...
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1answer
43 views

Breaking up Wiener processes with indicator functions?

Consider a Wiener process $W_t$ which is adapted to $\mathscr{F}_t$, where this filtration has all of the standard properties. I'm also working with a stock-standard probability space here. I want to ...
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1answer
288 views

Wiener martingale

Is $\frac{W^{2}(t)}{t}$ a martingale w.r.t. the usual filtration? $t>0$ and $W(t)$ is the Wiener process. What I have so far: Define $Z(t)=\frac{W^2(t)}{t}$. By Ito we get ...
4
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1answer
294 views

beginner's question about Brownian motion

I have just started learning about stochastic processes and I am confused with the notion of Brownian motion. The text defines (linear) Brownian motion under measure $\mathbb{P}$ as $B=(B_t; t\geq 0)$ ...
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2answers
101 views

Second order linear partial differential equation: $\partial_t u(t,x)+\frac12 \partial_{x,x} u(t,x)+u(t,x)v(x)=0$

Is there a way to solve $$ \partial_t u(t,x)+\frac12 \partial_{x,x}u(t,x)+u(t,x)v(x)=0? $$ This appeared as a condition for $$ X_t=u(t,B_t)e^{\int_0^tv(B_s)ds} $$ to be a martingale. With $B$ a ...
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1answer
78 views

Expection of Brownian Squared conditional on the end of the path

I have been asked as a brainteaser to compute the value of: $\mathbb{E}[W_t^2|W_T]$ with $t < T$ ? Does anyone know how to proceed ?
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1answer
1k views

Applying Ito formula to the Brownian bridge

Let $B$ be a standard Brownian motion and $$ W_t=(1-t)\int_0^t \frac{1}{1-s}dB_s $$ be a Brownian bridge. Calculate $dW_t$. To apply Ito formula define $$ f(t,B_t)=(1-t) \int_0^t\frac{1}{1-s}dB_s $$ ...
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1answer
392 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?
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1answer
450 views

Brownian motion question

Let $B(t)$ denote the standard Brownian motion and let $X(t)$ denote a Brownian motion with $X(0)=0$, drift $0$ and variance $9$. Find the distribution of $aB(s)+bB(t)$, where $a,b,s,t$ are real ...
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1answer
141 views

Differentiability of hitting time of Brownian motion

I have been trying to prove the following conjecture for a while, but so far to no avail. Would be very grateful for some tips! The conjecture is the following; Think of an $n$ dimensional Brownian ...
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1answer
336 views

Show that two random variables are equal in distribution

I found this exercise at the end of a chapter about Brownian motion. Let $(X_j)_{j=1}^{2^M}$ be independent standard Gaussian random variables, where $M$ is a integer. Define ...
3
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2answers
295 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
56 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} ...
5
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1answer
378 views

Brownian motion: changing the order of expectation and integration in $E \left( \int_s^t B_x dx \mid F_s \right)$

Let $B$ be a standard Brownian motion with induced filtration $F$. Is it true that, for $s<t$, $$ E \left( \int_s^t B_x dx \mid F_s \right) = \int_s^t E \left( B_x \mid F_s\right) dx \;? $$ To ...
2
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1answer
503 views

Arcsine law for Brownian motion

Here is the question: $(B_t,t\ge 0)$ is a standard brwonian motion, starting at $0$. $S_t=\sup_{0\le s\le t} B_s$. $T=\inf\{t\ge 0: B_t=S_1\}$. Show that $T$ follows the arcsinus law with ...
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1answer
191 views

local maximum of brownian motion

I have a question: Given two disjoint intervals $[a,b]$ and $[c,d]$, how to prove almost surely we have $$\sup_{t\in[a,b]}B_t\neq\sup_{t\in[c,d]}B_s$$ where $B$ is a standard brownian motion. I ...
2
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0answers
184 views

Correlated diffusion processes and covariance matrix

I'm really noob in maths topics so I hope you will excuse me if I use terms which aren't correct. I would like to simulate $n$ dimensional diffusion processes with $n$ noises. Each process has its ...
1
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1answer
108 views

Variable t times a Wiener Process W(1/t)

If $W(t)$ is a Wiener process and $V(t) = t\cdot W(1/t)$ is it possible to say that Since $W(1/t)\space \sim N(0,1/t)$ that $V(t) \sim t\cdot N(0,1/t)$? And if so then is $t\cdot N(0,1/t) = ...
3
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2answers
124 views

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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0answers
170 views

Show that $O_t$ is a Gaussian Process

Let $B_t$ be a Brownian motion process. Let $$O_t = e^{-\alpha t} \int^t_0 e^{\alpha s} dB_s$$ Find $\mathsf{E}[O_t]$ and show that $O_t$ is a Gaussian process. I think ...
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0answers
46 views

prove that two r.v.s share the same law

I have a question in my homework about Brownian motion. Does someone have a idea about the following question? Let $X=B^+$ or $|B|$ where $B$ is a standard BM, $p>1$ be a real number and $q$ its ...
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2answers
110 views

Certain transformation of Brownian motion is a submartingale

I have a question about a proof in Protter. Let $B$ Brownian motion and $u$ a harmonic (subharmonic) function. Then $u(B)$ is a local martingale (submartingale). I was able to show the case of local ...
2
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1answer
284 views

SDE - removal of the diffusion coefficients

I'm currently looking at stochastic differential equations with irregular coefficients such as $W^{1,p}_\mathrm{loc}$. If I have \begin{align} dX_t=b(X_t) \, dt+\sigma \, dW_t, \end{align} where $b\in ...
2
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1answer
86 views

A nonmeasurable set on $\mathbb{R}^{\left[0,\infty\right)}$.

Let $\left\{ X_{t}\right\} _{t\geq o}$ the canonical version of Brownian motion, i.e., if we consider $\Omega:=\mathbb{R}^{\left[0,\infty\right)}$ the set of the real valued functions on ...
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1answer
702 views

Brownian Motion and Reflection Principle

I am currently studying the Brownian motion and I am stuck with a problem related to the reflection principle. What I am trying to calculate is the probability that a standard Brownian Motion $X_t$ ...
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2answers
1k 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 ...
3
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1answer
170 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
173 views

Klenke's construction of Brownian motion

Why does Klenke's concise construction of Brownian motion via probability transition kernels satisfy the motion's characterizing properties, equations (14.17) and (14.18)? (results referenced in the ...
2
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3answers
121 views

Derivation of Wiener process first passage times using probability generating function?

I would like to find the distribution of first passage times in a simple Wiener process using the idea of probability generating functions. Thus there will be, at certain point, a limiting step to go ...
3
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1answer
974 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 ...
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2answers
220 views

Proof of Lévy's characerization of Brownian motion

There is a purely probability theoretical argument in the proof of Lévy's characterization of Brownian motion, which I do not completely understand. I think it is rather easy. Suppose we know ...
6
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2answers
423 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 ...
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1answer
105 views

Strong markov property on max of brownian motion

For $B_t$ Brownian Motion with drift $\mu<0$, I have the max value, $X = \max_{0<t<\infty}B_t$ . I need to prove with the Strong Markov Property that, $P(X>c+d)=P(X>c)P(X>d)$ a. It ...
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2answers
172 views

Max of Brownian motion with drift is finite almost surely

For $B_t$ Brownian Motion with drift $\mu<0$, I need to prove that the max value, $X = \max_{0<t<\infty}B_t$ is finite almost surely, ie $P(X<\infty)=1$. Now, I know that because the mean ...
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1answer
148 views

How to calculate this conditional probability

There's an equation in my script, which I do not understand. Let $(B_t)$ be a Brownian Motion and $\Gamma\in\mathcal{B}(\mathbb{R}^n)$, $t\ge s$ the equation is $$P(B_t\in\Gamma | ...
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1answer
389 views

Distribution of a stopping time

For $c\in\mathbb{R}$ we define the stopping time $\tau_c:=\inf\{t>0;X_t>c\}$ if $c\ge 0$ otherwise $\tau_c:=\inf\{t>0;X_t<c\}$. Let $X$ be a Markov Process on a metric space $E$ with ...
19
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2answers
535 views

Does Brownian motion visit every point uncountably many times?

Let $B_t$ be a one-dimensional standard Brownian motion. Is it true that, almost surely, for every $x \in \mathbb{R}$ the set $\{t : B_t = x\}$ is uncountable? Let $A_x$ be the event that $\{t : ...
2
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1answer
75 views

Covariation Paradox??

we can see that $\left\langle \int_0^t \! W_s \, \mathrm{d} s ,W_t \right\rangle_t = 0$ However if I am to use the expression $$\int_0^t \! W_s \, \mathrm{d} s= t W_t - \int_0^t \! s\, \mathrm{d} ...
2
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1answer
158 views

One correlated Stochastic Integral

If $${\rm Cov}[dW_t,dB_t]=\rho dt$$ then what is $$\mathbb{E} \left[\int_0^t\sigma_{1s}dW_s \int_0^t\sigma_{2s}dB_s\right]$$ where $\sigma_{1s}$ and $\sigma_{2s}$ are two deterministic functions ...
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1answer
456 views

calculate the conditional distribution of brownian motion

Suppose $W=(W_t)$ is a Brownian Motion with respect to a filtration $(\mathcal{F}_t)$. How can I compute the conditional distribution of $W_{t+h}$ given $\mathcal{F}_t$. I started like this: ...
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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 ...
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2answers
257 views

question about the bracket process of brownian motion

Suppose I have a multidimensional brownian motion $W=\{W_t\}$. Why is the following true: $$\langle W^k,W^l\rangle_t = \delta_{k,l}t$$ where $W^k$ denotes the k-th coordinate, $\langle ...
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1answer
2k views

Questions and Solutions in Brownian Motion and Stochastic Calculus?

I am currently studying Brownian Motion and Stochastic Calculus. I believe the best way to understand any subject well is to do as many questions as possible. Unfortunately, I haven't been able to ...
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3answers
1k views

Simulation of 2-dimensional Brownian motion

I am trying to simulate (for the first time) a 2-dimensional SDE, in Matlab. $$X(t)=F(t,X(t))\,dt + \sigma(t,X(t))\,dBt$$ I have no problem using the Euler-Maruyama method in the one dimensional ...
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1answer
212 views

The joint distribution of zeros of brownian motion

Let $\gamma_t$ be the last zero of brownian motion before $t$ and $\beta_t$ be the first zero after $t$. I need to calculate the joint distribution of $\gamma_t$ and $\beta_t$, i.e. $P(\gamma_t<x, ...
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2answers
746 views

Lottery “Sum” forecasting

I was wondering if anyone can provide some mathematical insights to forecasting the "SUM" in this link as a time series. It is an oscillatory, range bound and poisson distribution. How can Monte Carlo ...
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0answers
80 views

A Brownian motion starting from 0, it becomes -1 once reach -1, what is its expectation?

$X_t$ is a Brownian Motion, it reaches becomes -1 forever once it reaches -1. Mathematically, $T = \inf\{t: W_t = -1\}$ is a stopping time. When $t<T$, $X_t = W_t$ ,while $t\geq T$, $X_t = -1$. ...
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2answers
1k views

Hitting time of Brownian Motion with a drift

Let $X_t =x+bt+\sqrt{2}W_t$, where $W_t$ is a standard Brownian motion. Let $T=\inf\{t: |X_t|=1\}$. I am trying to find $\mathbb{E}[T]$ for the case $b\neq0$. Firstly, I am going to apply Girsanov to ...
1
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2answers
447 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 ...
3
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1answer
427 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: ...