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 $t-s$.
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, ...
2
votes
1answer
376 views
Absolute value of Brownian motion
I need to show that $$R_t=\frac{1}{|B_t|}$$ is bounded in $\mathcal{L^2}$ for $(t \ge 1)$, where $B_t$ is a 3-dimensional standard Brownian motion.
I am trying to find a bound for ...
2
votes
1answer
108 views
Doob's stopping time theorem with unbounded stopping time
Let $(X_t)_{t\geq0}$ be Brownan motion on $\mathbb R$, and $\tau$ is a stopping time adapted with the natural filtration generated by the Brownian motion. If $X_0=0$, $E(e^{\tau/2})<+\infty$.
...
2
votes
1answer
65 views
Distribution of the integral of a diffusion process
Suppose $X(t)$ is a diffusion process with $E[X(t)]=0$ and variances $\sigma^2_t$ concave in time. If $X$ is also a Brownian motion, then the distribution of $\int_0^T X(t) dt$ is known to be ...
2
votes
1answer
76 views
Independent increments of $X_t:=\int_0^t\phi(s) dW_s$
Motivated through the following question
Can we prove directly that $M_t$ is a martingale, I want to ask this in a separate question. Suppose we have a deterministic function $\phi$ which belongs to ...
2
votes
1answer
71 views
Show that $M_t = \int_0 ^t \exp{((B_2(s)^2)} dB_1(s)$ is not a continuous square integrable martingale
Consider the following $\mathcal F_t$- (continouous) local martingale $$M_t = \int_0 ^t \exp{((B_2(s)^2)} dB_1(s)$$
where $\left(B_t\right)_{t\geq0} =\left(B_1(t),B_2(t)\right)_{t\geq0}$ is ...
2
votes
1answer
84 views
What is the conditional distribution of $B(s)\mid B(t_1)=x_1,B(t_2)=x_2$ for $0<t_1<s<t_2$?
Given that $\{B_t,t\ge0\}$ is a standard Brownian process. What is the conditional distribution of $B(s)$ given $B(t_1)=x_1$ and $B(t_2)=x_2$, for $0<t_1<s<t_2$?
My try: First i tried to ...
2
votes
1answer
69 views
Applying Ergodic Theorem on fractional Brownian motion
For a fractional Brownian motion $B_H$ consider the sequence for $p>0$
$$Y_{n,p}={1\over n}\sum\limits_{i=1}^n \left|B_H(i)-B_H(i-1)\right|^p.$$
By the Ergodic Theorem it is ...
2
votes
2answers
97 views
Expectation of a stopping time of a Wiener process
How can we calculate $\mathbb{E}(\tau)$ when $\tau=\inf\{t\geq0:B^2_t=1-t\}$?
If we can prove that $\tau$ is bounded a.s. (i.e. $\mathbb{E}[\tau]<\infty$), then we can use the fact that ...
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 ...
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 ...
2
votes
1answer
111 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
$$P(B_t\in\Gamma | ...
2
votes
1answer
63 views
Covariation Paradox??
we can see that $\left\langle \int_0^t \! W_s \, \mathrm{d} s ,W_t \right\rangle_t = 0$
However if I am to use the expression
$$\int_0^t \! W_s \, \mathrm{d} s= t W_t - \int_0^t \! s\, \mathrm{d} ...
2
votes
1answer
141 views
Stopping time on Wiener Process
Let $W_t$ be a Wiener process and for $a\geq0$
$$\tau_a:=\inf \left\{ t\geq0: |W_t|=\sqrt{at+7} \right\}.$$
Is $\tau_a<\infty$ almost everywhere? What about $E(\tau_a)$ then?
2
votes
1answer
291 views
Difference between a Brownian Motion and the root of its square
Let $W_{t}$ be a Wiener Process (a Brownian Motion starting at $W_{0} = 0$). What is the difference between $W_{t}$ and $\sqrt{W_{t}^{2}}$?
Using the Ito formula (in differential notation), ...
2
votes
1answer
238 views
Expectation of an integral of the minimum of a Brownian motion and a constant
I would like to compute the expectation of the following expectation
$\mathbb{E}[\int_a^\infty e^{-rt}\min(x_t,c)\,dt]\,$
where a, r, c are constants, $dx_t = \mu x_t dt + \sigma x_t dW_t$ is a ...
2
votes
1answer
89 views
Brownian Motion Conditional Expectation Question
I have a real number $x$, and $W$ is a standard Brownian motion. Let $0 < s < t$. How to find
$$
\mathsf E[W_s | W_t = x]
$$
Please provide me with a step by step answer as I want to ...
2
votes
0answers
43 views
Negative moments of a functional of Wiener process
At the moment I am reading D. Nualart's The Malliavin Calculus and Related Topics. The problem I am trying to solve is the following: Show that the random variable $\int_0^1 s^2\arctan W_s\, ds$ ...
2
votes
1answer
41 views
Existence Brownian Motion
I'm reading through a proof of the existence of a Brownian motion and at some point they state that for $0\leq t_{0}<t_{1}...<t_{n}$ there exist multivariate normal distributions with covariance ...
2
votes
0answers
57 views
Independence of Brownian Motion with respect to a stopping time
Let $B_t$ be a brownian motion, $B_0=0$, and $\gamma \in \mathbb{R}$.
Now, let's build the following stopping time:
\begin{equation}
T = \inf \{ t \geq 0 : |B_t + \gamma t| = 1 \}.
\end{equation}
If ...
2
votes
0answers
77 views
A problem with regard to Wiener process
Let $W$ be a Wiener process and $U_x$ is the amount of time spent below $x$ during time interval $(0,1)$. Hence $U_x=\int\limits_0^1I_{\{W(t)<x\}}dt$. My question is: what is the probability ...
2
votes
1answer
46 views
Clarke Ocone representation formula
Let $(B_t)_{t}$ a Brownian motion and $F \in L^2(\Omega,\mathcal{F}_T,\mathbb{P})$. Then we know by Itô's representation theorem that there exist a process $X$ such that $$F=\mathbb{E}F+\int_0^T X_s ...
2
votes
0answers
77 views
Integral representation of fractional Brownian motion
Let $H\in$ $]0,1[$. A fractional Brownian motion $\left(B_H(t)\right)_{t\geq 0}$ can be represented as $${1\over C(H)}\int_\mathbb{R}\left((t-s)_+^{H-{1\over2}}-(-s)_+^{H-{1\over2}}\right)dB(s)$$ ...
2
votes
1answer
82 views
expectation of a process of a multidimensional brownian motion
Let $B(t)=(B_{1}(t),B_{2}(t),B_{3}(t))$ be a standard three dimensional Brownian motion
(i.e. it has independent components and starts at the origin). Now let $a=(a_{1},a_{2},a_{3})\neq(0,0,0)$ be a ...
2
votes
1answer
329 views
Transition density and distribution: (Ornstein–Uhlenbeck process)
Let $\left(X_{t},\, t\geq0\right)$ be the weak solution to the SDE
below with $\alpha,\,\beta,\,\gamma$ constants:
$$
dX_{t}=(-\alpha X_{t}+\gamma)dt+\beta dB_{t}\quad\forall t\geq0,\, X_{0}=x_{0}
$$
...
2
votes
1answer
113 views
About Brownian motion
Let $T$ be the last time before $1$ a Brownian motion visits $0$. Explain why
$$X(t)=B(t+T)-B(T)=B(t+T)$$ is not a Brownian motion. This problem is from Introduction to Stochastic Calculus with ...
2
votes
1answer
73 views
Integrating a Brownian Bridge conditioned above a linear boundary
The Setup:
A Brownian Bridge $B$ is a Brownian Motion on time interval $[0, 1]$ conditioned such that $B(0) = B(1) = 0$. I have a function $f(t) = mt+b$ with $m, b$ set such that $C(t) \le 0$ for $t ...
2
votes
0answers
103 views
Correlated diffusion processes and covariance matrix
I'm really noob in maths topics so I hope you will excuse me if I use terms which aren't correct.
I would like to simulate $n$ dimensional diffusion processes with $n$ noises.
Each process has its ...
2
votes
0answers
116 views
Show that $O_t$ is a Gaussian Process
Let $B_t$ be a Brownian motion process. Let $$O_t = e^{-\alpha t} \int^t_0 e^{\alpha s} dB_s$$ Find $\mathsf{E}[O_t]$ and show that $O_t$ is a Gaussian process.
I think ...
2
votes
1answer
121 views
SDE - removal of the diffusion coefficients
I'm currently looking at stochastic differential equations with irregular coefficients such as $W^{1,p}_\mathrm{loc}$. If I have
\begin{align}
dX_t=b(X_t) \, dt+\sigma \, dW_t,
\end{align}
where $b\in ...
2
votes
0answers
99 views
Ruin probability
Let $X_t$ be a solution of the stochastic differential equation
$$ dX_t= -\frac{c-1}{2 X_t}dt+ dB_t, \, \qquad X_0=x_0$$
where $c$ is a real constant and $B_t$ is a Brownian motion. Can you give me ...
1
vote
1answer
171 views
How do you show this is a martingale?
How do you show the following process is a martingale? My notes say it is a martingale by I can't work it out.
$$
E[e^{\sigma B(t) - \frac{\sigma ^2 t}{2}} | \mathscr{F}(s)]
$$
I tried to multiply ...
1
vote
1answer
129 views
Expectation of Stopping Time w.r.t a Brownian Motion
How do you take the expectation of a stopping time with respect to a Brownian motion? The specific question is:
$$
\tau = \inf\{ t \ge 0: B(t) \in \{-a, b\}\}
$$
I understand the optional stopping ...
1
vote
1answer
50 views
Backward martingale property of quadratic variation
Let $\pi_n$ denotes a refining sequence of partitions of a finite closed interval (refining means $\pi_n\subset\pi_{n+1}).$ And we denote $\pi_n B = \sum_{t_i\in \pi_n}(B_{t_{i+1}}-B_{t_i})^2$, where ...
1
vote
1answer
49 views
How to prove that for Brownian motion in $(a, b)$ $\mathbb{E}^x[\min(H_a, H_b)] = (x-a)(b-x)$?
i'm wondering if anyone can help me with proving the fact that for BM in the interval $(a,b)$ and with $$H_y = \inf\{t>0: X_t = y\},$$ the following is true: $$\mathbb{E}^x[\min(H_a, H_b)] = ...
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:
$$
\int_0^t d(e^{-ut}X(s)) = X(0) + \sigma \int_0^t e^{-us}dB(s)
...
1
vote
2answers
184 views
Derivation of an alternative representation of the Ornstein-Uhlenbeck process
The Ornstein-Uhlenbeck process can be defined as:
$X_t = e^{-\lambda t} \left( X_0 + \int_0^t e^{\lambda s} dB_s \right)$
where $\lambda > 0$ and $\{ B_t \}_{t \geq 0}$ is the standard Brownian ...
1
vote
1answer
166 views
Continuity of the Ito integral
Let $B_t$ be a 1-dimensional Brownian motion. I am following "Stochastic Differential Equations" by Bernt Øksendal. On the page 32 (it is displayed in the link I've put) there is a proof of existence ...
1
vote
2answers
68 views
Standard Brownian Motion
Let $\{X_t,t\ge 0\}$ be a standard Brownian motion. Compute the density of $X_t$ conditioned by $X_{t_1}$ and $X_{t_2}$ assuming that $t_1 <t<t_2$.
Can anyone give me some hint to start the ...
1
vote
3answers
34 views
Scaling and time inversion for Brownian motion basically the same?
Let $B(t)$ be a Brownian motion. For $a>0$, we have the scaling relation
$$\hat{B}(t)=aB(t/a^2) \sim B(t)$$
and $\hat{B}(t)$ is also a Brownian motion.
The time inversion formula states that
...
1
vote
2answers
108 views
variance of square of brownian motion increment
In other words,
$$\text{Var}\left\{ [W(t) - W(s)]^2 \right\} = \mathbb E \left\{ (W(t) - W(s))^4 \right\} - \left[ E\left\{ (W(t) - W(s))^2 \right\} \right]^2 $$
How is this equal to $(t-s)^2$ ...
1
vote
1answer
78 views
Running maximum of Wiener process
The joint distribution of the running maximum
$ M_t = \max_{0 \leq s \leq t} W_s $
and $W_t$ is
$f_{M_t,W_t}(m,w) = \frac{2 ( 2 m - w)}{t\sqrt{2 \pi t}}e^{-\frac{(2m-w)^2}{2t}}, m ...
1
vote
1answer
43 views
Convergence in $L^{2}(\Omega)$
Let $T>0$ and $P^{n}:=\lbrace0=t_{0}^{n}<t_{1}^{n}<...<t_{m_{n}}^{n}=T\rbrace$ be the $n$- th division of the interval $[0,T]$ such that $\delta(P^{n})\to0$, as $n\to\infty$, where ...
1
vote
2answers
44 views
The probability of a Brownian particle traveling a distance $L$ before returning to its point-of-origin
What's the probability that a Brownian particle diffusing along a one-dimensional interval returns to its point of origin before traveling a distance $L$?
We know that in the limit of a random walk ...
1
vote
1answer
122 views
d-dimensional Brownian motion and martingales
I was solving questions from the Martingales chapter in "Stochastic Processes" by Richard Bass. There was a question regarding d- dimensional Brownian motions(BM):
Let $(W_t^1,...,W_t^d)$ be a d ...
1
vote
1answer
26 views
a homework question about Levy air
I have a question in my homework:
Let $X_t$ and $Y_t$ be two Brownian motions issue de $0$ and define
$$S_t=\int_0^tX_s\,dY_s-\int_0^tY_s\,dX_s$$
Show that
$$E[e^{i\lambda S_t}]=E[\cos(\lambda ...
1
vote
1answer
50 views
What is the distribution of $B(s) + B(t)$ , $s \leq t$? Answer already given. Just curious of a process within the answer.
This is question is under the topic of Brownian motion.
The question is:
What is the distribution of $X(s) + X(t)$, when $s \leq t?$
Answer:
$$X(s) + X(t) = 2X(s) + X(t) − X(s)$$
Now $2X(s)$ ...
1
vote
2answers
70 views
Independence of Brownian motion
While by definition the increments of a Brownian motion are independent, it is unclear to me whether (that implies that) the random variables $W_t$ and $W_s$ are independent for $t \neq s$. While ...
1
vote
1answer
40 views
Intuition for Rough path
I pray to kindly show the intuition behind the concept of rough path. Google provided some links that deal with the notion of rough path but was difficult for me to have an idea.
1
vote
1answer
174 views
Integration of Wiener process: $\int_{t_1}^{t_2} dB(s)$
We all know that $\int_0^t dB(s) = B(t)$, where $B(t)$ is a standard Brownian Motion.
However, is the following identity true? Also, why or why not?
$\boxed{ \displaystyle \ \ \int_{t_1}^{t_2} ...
