Question related to Brownian motion, a stochastic process denoted $W_t$, $t\geq 0$, with independent increments, such that $W(t)-W(s)$ is normally distributed, with $0$ mean and variance $t-s$.
0
votes
1answer
185 views
Brownian Motion(symmetry, time reversal and scaling)
How do I prove the symmetry of Brownian motion? ( -w(t) is a Brownian motion?)? Also i read in many places about time reversal and scaling of brownian motions as prepositions. I would like to learn ...
2
votes
1answer
73 views
Integrating a Brownian Bridge conditioned above a linear boundary
The Setup:
A Brownian Bridge $B$ is a Brownian Motion on time interval $[0, 1]$ conditioned such that $B(0) = B(1) = 0$. I have a function $f(t) = mt+b$ with $m, b$ set such that $C(t) \le 0$ for $t ...
2
votes
1answer
216 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. ...
1
vote
1answer
28 views
Integrating $d(e^{-ut}X(t))$, where $X(t)$ is stochastic.
Given that $\sigma e^{-ut}dB(t) = d(e^{-ut}X(t))$, where $X(t)$ is a stochastic process and $B(t)$ is a Wiener process, we have that:
$$
\int_0^t d(e^{-ut}X(s)) = X(0) + \sigma \int_0^t e^{-us}dB(s)
...
1
vote
1answer
54 views
Derive an SDE for $B^2(t)$, where $B(t)$ is standard Brownian Motion
Original Question:
Derive an SDE for $B^2(t)$, where $B(t)$ is standard Brownian Motion.
Attempt at an answer:
Apply Ito's calculus over $f(t,b):= B^2(t)$.
$$df(t,b) = \frac{\partial ...
1
vote
1answer
117 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 ...
0
votes
1answer
36 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 ...
1
vote
1answer
105 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
...
2
votes
1answer
120 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 $B=(B_t; t\geq 0)$ ...
2
votes
2answers
82 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 ...
0
votes
1answer
65 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 ?
0
votes
1answer
301 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
$$
f(t,B_t)=(1-t) \int_0^t\frac{1}{1-s}dB_s
$$
...
3
votes
1answer
162 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?
1
vote
1answer
125 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 ...
1
vote
1answer
86 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 ...
2
votes
1answer
148 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
votes
1answer
40 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
$$
S_n=\sum_{i=0}^{n-1} ...
4
votes
1answer
109 views
Brownian motion: changing the order of expectation and integration in $E \left( \int_s^t B_x dx \mid F_s \right)$
Let $B$ be a standard Brownian motion with induced filtration $F$.
Is it true that, for $s<t$,
$$
E \left( \int_s^t B_x dx \mid F_s \right) =
\int_s^t E \left( B_x \mid F_s\right) dx \;?
$$
To ...
1
vote
1answer
146 views
Arcsine law for Brownian motion
Here is the question:
$(B_t,t\ge 0)$ is a standard brwonian motion, starting at $0$.
$S_t=\sup_{0\le s\le t} B_s$. $T=\inf\{t\ge 0: B_t=S_1\}$.
Show that $T$ follows the arcsinus law with ...
1
vote
1answer
64 views
local maximum of brownian motion
I have a question:
Given two disjoint intervals $[a,b]$ and $[c,d]$, how to prove almost surely we have
$$\sup_{t\in[a,b]}B_t\neq\sup_{t\in[c,d]}B_s$$
where $B$ is a standard brownian motion. I ...
2
votes
0answers
103 views
Correlated diffusion processes and covariance matrix
I'm really noob in maths topics so I hope you will excuse me if I use terms which aren't correct.
I would like to simulate $n$ dimensional diffusion processes with $n$ noises.
Each process has its ...
1
vote
1answer
71 views
Variable t times a Wiener Process W(1/t)
If $W(t)$ is a Wiener process and $V(t) = t\cdot W(1/t)$ is it possible to say that
Since $W(1/t)\space \sim N(0,1/t)$
that $V(t) \sim t\cdot N(0,1/t)$?
And if so then is $t\cdot N(0,1/t) = ...
3
votes
2answers
83 views
Brownian motion: Show $\lim \sum W_{i} (W_{i+1}-W_{i})=\frac12 W^2_t-\frac12 t$ in probability.
Let $\{t_i\}_{i=1}^n$ be a partition of $[0,t]$ and $W$ a standard Brownian motion. Write $W_i$ for $W_{t_i}$.
Show
$$
\lim \sum W_{i} (W_{i+1}-W_i)=\frac12 W^2_t-\frac12 t
$$
where the limit is in ...
2
votes
0answers
116 views
Show that $O_t$ is a Gaussian Process
Let $B_t$ be a Brownian motion process. Let $$O_t = e^{-\alpha t} \int^t_0 e^{\alpha s} dB_s$$ Find $\mathsf{E}[O_t]$ and show that $O_t$ is a Gaussian process.
I think ...
1
vote
0answers
37 views
prove that two r.v.s share the same law
I have a question in my homework about Brownian motion. Does someone have a idea about the following question?
Let $X=B^+$ or $|B|$ where $B$ is a standard BM, $p>1$ be a real number and $q$ its ...
2
votes
2answers
61 views
Certain transformation of Brownian motion is a submartingale
I have a question about a proof in Protter. Let $B$ Brownian motion and $u$ a harmonic (subharmonic) function. Then $u(B)$ is a local martingale (submartingale). I was able to show the case of local ...
2
votes
1answer
120 views
SDE - removal of the diffusion coefficients
I'm currently looking at stochastic differential equations with irregular coefficients such as $W^{1,p}_\mathrm{loc}$. If I have
\begin{align}
dX_t=b(X_t) \, dt+\sigma \, dW_t,
\end{align}
where $b\in ...
2
votes
1answer
70 views
A nonmeasurable set on $\mathbb{R}^{\left[0,\infty\right)}$.
Let $\left\{ X_{t}\right\} _{t\geq o}$ the canonical version of Brownian
motion, i.e., if we consider $\Omega:=\mathbb{R}^{\left[0,\infty\right)}$
the set of the real valued functions on ...
0
votes
1answer
381 views
Brownian Motion and Reflection Principle
I am currently studying the Brownian motion and I am stuck with a problem related to the reflection principle.
What I am trying to calculate is the probability that a standard Brownian Motion $X_t$ ...
4
votes
2answers
372 views
Prove the time inversion formula is brownian motion
Let $B=(B_t)_{t\geq 0}$ be a brownian motion. Show the time inversion formula $\hat{B}=(B_t)_t\geq0$ is a brownian motion, where for $t \geq 0$ we set $\hat{B}=0$ for $t=0$ and $\hat{B}=tB_{1/t}$ for ...
3
votes
1answer
108 views
Probability for brownian motion
How can I prove it?
For $b>a>0$, show that
$$
\operatorname{Pr}\left({\sup_{t\geqslant 0}\left(\frac{b+X(t)}{1+t}\right)\geqslant a}\right)=e^{-2a(a-b)}
$$
where $X(t)$ is a Brownian ...
3
votes
1answer
156 views
Klenke's construction of Brownian motion
Why does Klenke's concise construction of Brownian motion via probability transition kernels satisfy the motion's characterizing properties, equations (14.17) and (14.18)?
(results referenced in the ...
2
votes
3answers
87 views
Derivation of Wiener process first passage times using probability generating function?
I would like to find the distribution of first passage times in a simple Wiener process using the idea of probability generating functions. Thus there will be, at certain point, a limiting step to go ...
4
votes
1answer
246 views
To show that a given process is Gaussian
Suppose I have given a Brownian Motion $W$, this is a Gaussian process, and I define:
$$B_s:=W_{t-s}-W_t$$
for $0\le s\le t$. Clearly this random variable has expectation zero. For the covariance ...
1
vote
2answers
120 views
Proof of Lévy's characerization of Brownian motion
There is a purely probability theoretical argument in the proof of Lévy's characterization of Brownian motion, which I do not completely understand. I think it is rather easy. Suppose we know
...
5
votes
2answers
235 views
Laplace transform of integrated geometric Brownian motion
Is there any closed form of the Laplace transform of an integrated geometric Brownian motion ?
A geometric Brownian motion $X=(X_t)_{t \geq 0}$ satisifies $dX_t = \sigma X_t \, dW_t$ where ...
0
votes
1answer
65 views
Strong markov property on max of brownian motion
For $B_t$ Brownian Motion with drift $\mu<0$, I have the max value, $X = \max_{0<t<\infty}B_t$ .
I need to prove with the Strong Markov Property that, $P(X>c+d)=P(X>c)P(X>d)$
a. It ...
0
votes
2answers
108 views
Max of Brownian motion with drift is finite almost surely
For $B_t$ Brownian Motion with drift $\mu<0$, I need to prove that the max value, $X = \max_{0<t<\infty}B_t$ is finite almost surely, ie $P(X<\infty)=1$.
Now, I know that because the mean ...
2
votes
1answer
110 views
How to calculate this conditional probability
There's an equation in my script, which I do not understand. Let $(B_t)$ be a Brownian Motion and $\Gamma\in\mathcal{B}(\mathbb{R}^n)$, $t\ge s$ the equation is
$$P(B_t\in\Gamma | ...
2
votes
1answer
151 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 ...
17
votes
2answers
268 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
votes
1answer
63 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
$$\int_0^t \! W_s \, \mathrm{d} s= t W_t - \int_0^t \! s\, \mathrm{d} ...
2
votes
1answer
86 views
One correlated Stochastic Integral
If
$${\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]$$
where $\sigma_{1s}$ and $\sigma_{2s}$ are two deterministic functions ...
0
votes
0answers
51 views
Controlling auto-correlated 1D Brownian motion
I have 1D Brownian motion process $x(t)$, and ability to control it.
The control allows to shift the $x$ by $D$ at any time.
I need the controlled process to be zero-mean, and to use the control ...
1
vote
1answer
246 views
calculate the conditional distribution of brownian motion
Suppose $W=(W_t)$ is a Brownian Motion with respect to a filtration $(\mathcal{F}_t)$. How can I compute the conditional distribution of $W_{t+h}$ given $\mathcal{F}_t$.
I started like this: ...
4
votes
1answer
363 views
Expectation value of a product of an Ito integral and a function of a Brownian motion
this problem has come up in my research and is confusing me immensely, any light you can shed would be deeply appreciated.
Let $B(t)$ denote a standard Brownian motion (Wiener process), such that the ...
1
vote
2answers
116 views
question about the bracket process of brownian motion
Suppose I have a multidimensional brownian motion $W=\{W_t\}$. Why is the following true:
$$\langle W^k,W^l\rangle_t = \delta_{k,l}t$$
where $W^k$ denotes the k-th coordinate, $\langle ...
1
vote
1answer
409 views
Questions and Solutions in Brownian Motion and Stochastic Calculus?
I am currently studying Brownian Motion and Stochastic Calculus. I believe the best way to understand any subject well is to do as many questions as possible. Unfortunately, I haven't been able to ...
0
votes
3answers
396 views
Simulation of 2-dimensional Brownian motion
I am trying to simulate (for the first time) a 2-dimensional SDE, in Matlab.
$$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 ...
2
votes
1answer
152 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, ...
