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 $t-s$.
1
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0answers
31 views
Independence of Brownian motion-related stopping times
Let $(B_t,\mathcal{F}_t)_{t \geq 0}$ a Brownian motion on a probability space $(\Omega,\mathcal{A},\mathbb{P})$. For $a \in \mathbb{R}$ define a stopping time $\tau_a$ by $$\tau_a := \tau(a) := ...
0
votes
0answers
74 views
How to prove this inequality involving integration with respect to Brownian motion?
If $B_t$ is the Brownian Motion, I have to verify that
$$E\left\lvert\int_s^t G(t,w)\,dB_t\right\rvert^6\leq 15^2\cdot (t-s)^2\cdot\int_s^t E\lvert G(t,w)\rvert^6\,dt$$
1
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2answers
103 views
variance of square of brownian motion increment
In other words,
$$\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 $$
How is this equal to $(t-s)^2$ ...
2
votes
1answer
65 views
Distribution of the integral of a diffusion process
Suppose $X(t)$ is a diffusion process with $E[X(t)]=0$ and variances $\sigma^2_t$ concave in time. If $X$ is also a Brownian motion, then the distribution of $\int_0^T X(t) dt$ is known to be ...
1
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1answer
49 views
How to prove that for Brownian motion in $(a, b)$ $\mathbb{E}^x[\min(H_a, H_b)] = (x-a)(b-x)$?
i'm wondering if anyone can help me with proving the fact that for BM in the interval $(a,b)$ and with $$H_y = \inf\{t>0: X_t = y\},$$ the following is true: $$\mathbb{E}^x[\min(H_a, H_b)] = ...
3
votes
1answer
56 views
Limit of occupation times for Brownian motion
Let $B_t$ be a standard Brownian motion on $\mathbb R$ started at $0$. For $A\subset\mathbb R$ Lebesgue measurable, let $\mu_T(A) = \frac{1}{T} m(t \leq T: B_t \in A)$, where $m$ is Lebesgue measure. ...
1
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1answer
77 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 ...
2
votes
1answer
75 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 ...
0
votes
1answer
39 views
Variance for the distance between two Brownian particles vs. a Brownian particle and a stationary particle
I have two Brownian particles, $B_1$ and $B_2$ (with diffusion coefficients $D_1$ and $D_2$), at coordinates $P_1$ and $P_2$ in a three-dimensional fluid.
I let the system evolve for $t$ seconds. ...
2
votes
1answer
29 views
Motion of the centroid of $k$ Brownian particles?
Imagine we have $k$ Brownian particles diffusing in a three-dimensional solution, where each particle has the same diffusion coefficient $D$ (measured in $\mu^2/sec$). Now imagine that we have a ...
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
1answer
66 views
Convergence to Brownian motion integral
Let $X_i$ be i.i.d with $\mathbb{E}(X_i) = 0$ and $Var(X_i) =1, \, S_n = \sum_{i=1}^n X_i$. I would like to show that $\sum_{i=1}^n \frac{f(S_i/\sqrt{n})}{n}$ converges to $\int_0^1 f(B_t)dt$ in ...
1
vote
2answers
43 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 ...
3
votes
1answer
92 views
Brownian Bridge Representation
Let $B_t$ be a Wiener Process, then $U_t=B_t-tB_1,~0\le t \le 1$ is a Brownian bridge.
Show that $X_t=(1+t)U_{{t}/({1+t})}$ is a Wiener Process. I'm not quite sure how to start this off.
Any help ...
3
votes
1answer
108 views
linear combination of two Wiener processes
I have a question concerning the linear combination of two Wiener processes (please see http://en.wikipedia.org/wiki/Wiener_process for a definition). Let $W$ and $\tilde{W}$ be two Wiener processes ...
1
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0answers
27 views
Simulating of GBM
I have a question regarding the simulation of a GBM. I have found similar questions here but nothing which takes reference to my specific problem:
Given a GBM of the form
$dS(t) = \mu S(t) dt + ...
3
votes
1answer
34 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 ...
0
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1answer
93 views
Is the following a Wiener process?
This is a worked example on Wiener processes.
Question: Pick a normally distributed random variable $Z \sim N(0,1)$, then define $W(t) = Z\sqrt{t}$. Is $W(t)$ a Wiener process?
Answer:
...
3
votes
2answers
103 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 ...
1
vote
1answer
54 views
Absolute continuity of the distribution of $X_t=aB_t+bt$, $Y_t=a(t)B_t$ with respect to the Wiener measure
Let
$B_t:$ 1-dimensional Brownian motion, $P:$ its distribution on the Wiener space $C([0,1],\mathbb{R})$
$X_t=aB_t+bt\text{; }t \in [0,1]$, $P_{a,b}$ its distribution
$Y_t=a(t)B_t\text{; }a:[0,1] ...
5
votes
1answer
207 views
Computing the limit of the expectation of a function of a stochastic process (phew!)
I state my problem in a few lines then describe what I have already done.
I have a quite simple stochastic differential equation (SDE):
$dx=-2x \, dt+\sqrt{1-x^2} \, dW$ with $W$ a brownian.
I ...
3
votes
1answer
46 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 ...
1
vote
1answer
121 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
29 views
Cost for hedges under a Wiener process
I'm trying to estimate the hedging costs relating to a financial derivative which moves like a Wiener process, and I'm struggling to find the correct setup to solve the problem.
Suppose I have a ...
5
votes
1answer
61 views
How to show that the following process is a submartingale
Suppose we have a filtration $(\mathcal{F}_t)$ satisfying the usual conditions. Let $W$ be a Brownian Motion with respect to that filtration. We define the two processes
$X_t:=W^2_t$ and ...
3
votes
1answer
93 views
Fixed-Time Brownian Motion Exit Probabilities
A standard computation using martingale techniques allows us to compute probability that a Brownian motion started at zero exits the interval $[-a,b]$ ($a, b > 0$) at $-a$ or $b$. It appears to me ...
0
votes
0answers
41 views
Probability of a brownian motion leaving some area
Let $B_t$, $t\geq0$ be a standard $n$-dimensional Brownian motion, that is $B_t(\omega)\in\mathbb{R}^n$ and let $\Lambda\subset\mathbb{R}^n$ be some ball such that the Brownian motion starts within ...
3
votes
1answer
99 views
convergence ito integral
It is easy to calculate the integral $\int_0^T B_t \, dB_t=\frac{1}{2}B_T^2-\frac{1}{2}T$
That means I showed that $\int_0^T S_n \, ...
0
votes
1answer
30 views
How can we easily compute $\mathbb{E} [ \left|W_t\right| ^\alpha]$?
How can we easily compute $\mathbb{E} [ \left|W_t\right| ^\alpha]$, where $\alpha \in \mathbb R^*_+ $ and $W = (W_t)_{t \geq 0}$ is the one dimensional standard Brownian motion (or wiener process)?
6
votes
1answer
269 views
Expectation of an integral w.r.t. Brownian Motion
I know the following statement:
if $f$ is a deterministic function and continuous, i.e. $f\in C^0([0,T],\mathbb{R})$, then $\int f(s)dW_s$ is normally distributed with mean zero and variance $\int ...
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 ...
1
vote
1answer
52 views
Fractional Brownian motion as integral, mean zero
Let $H\in$ $]0,1[$. A fractional Brownian motion $\left(B_H(t)\right)_{t\geq 0}$ can be represented as $$X(t)={1\over ...
2
votes
1answer
91 views
Translational invariance of Brownian motion
Let $(\Omega,\mathcal{A},\mathbb{P})$ a probability space, $(X_t,\mathcal{F}_t)_{t \geq 0}$ a time-homogeneous Markov process. A paper I read defines a probability measure $\mathbb{P}^x$ by
...
3
votes
1answer
83 views
Brownian motion interesting question
I found this interesting question on the internet, but unfortunately I could not solve it.
What is probability that Brownian motion (starting at origin) has value 1 before having value -2?
3
votes
1answer
75 views
Brownian motion and hitting frequency
Suppose we have a Brownian motion $B_t$ with $B_0 = 0$ and $B_t - B_s \sim N(0,t-s)$. Every time $B_t$ hits $\pm h$, where $h$ is some "barrier" $>0$, I pay someone £1 and the brownian motion ...
0
votes
1answer
77 views
How to show that $X_t = \frac{1}{\left| B_t -x\right|}\mathbb{1}_{\left\{ B_t \neq x\right\}}$ (“inverse brownian”) is a martingale?
Consider
$$X_t = \frac{1}{\left| B_t -x\right|}\mathbb{1}_{\left\{ B_t \neq x\right\}}$$
where $ \left(B_{t }\right)_{t \geq 0}$ is a $ \mathcal F_t$- brownian motion in $\mathbb R ^3$, null at ...
2
votes
1answer
71 views
Show that $M_t = \int_0 ^t \exp{((B_2(s)^2)} dB_1(s)$ is not a continuous square integrable martingale
Consider the following $\mathcal F_t$- (continouous) local martingale $$M_t = \int_0 ^t \exp{((B_2(s)^2)} dB_1(s)$$
where $\left(B_t\right)_{t\geq0} =\left(B_1(t),B_2(t)\right)_{t\geq0}$ is ...
-3
votes
1answer
165 views
How to prove the martingale?
How to prove that the integral $\int_{0}^{+\infty}\upsilon e^{-ru}S_{u}dW_{u}^{Q}$ is a martingale under Q where $S_{t}$ is a martingale under Q and $\mathbb{E}^{Q}[\int_{0}^{+\infty}|\upsilon ...
3
votes
1answer
78 views
Autocorrelation of wrapped Wiener process
Let $\phi(t)$ be a Brownian Walk (Wiener Process), where $\phi\in[0,2\pi)$. As such we work with the variable $z(t)=e^{i\phi(t)}$. I would like to calculate
$$E(z(t)z(t+\tau)).$$
This is equal to ...
7
votes
1answer
163 views
Integral of the positive part of a Brownian motion
Let $X(t)$ be the standard Brownian motion, I need to find the distribution of $S=\int_{0}^T(X(t))^+dt$, where $(x)^+=\max\{0,x\}$.
I want to use the distribution to get a concentration bound for ...
4
votes
1answer
138 views
Brownian Motion Covariance: max instead of min
It is known that $\operatorname{Cov}(B_t,B_s)=\min(t,s)$ where $B$ is Brownian motion.
Can one think of an Ito process or integral (preferrably plain Gaussian process) $W$ such that ...
4
votes
1answer
185 views
Covariance of Brownian-motion-like processes
We know that $\operatorname{Cov}(B_s,B_t)=\min(s,t)$ if $B_t$ is Brownian motion.
What is $\operatorname{Cov}(B_{f(s)},B_{f(t)})$ for some injective $f$?
How can I write $B_{f(t)}$ in an Ito ...
3
votes
1answer
438 views
Covariance of Brownian Bridge?
I am confused by this question. We all know that Brownian Bridge can also be expressed as:
$$Y_t=bt+(1−t)\int_a^b \! \frac{1}{1-s} \, \mathrm{d} B_s $$
Where the Brownian motion will end at b at $t ...
2
votes
2answers
97 views
Relation between $\text{d}M_t$ and $\text{d}B_t$ when $M_t=\max_{0\leq s\leq t}B_s$
Let $B_t$ be a standard Wiener motion. What can we say about $\text{d}M_t$ and $\text{d}B_t$ when $M_t=\max_{0\leq s\leq t}B_s$? Is there a relation?
2
votes
1answer
81 views
What is the conditional distribution of $B(s)\mid B(t_1)=x_1,B(t_2)=x_2$ for $0<t_1<s<t_2$?
Given that $\{B_t,t\ge0\}$ is a standard Brownian process. What is the conditional distribution of $B(s)$ given $B(t_1)=x_1$ and $B(t_2)=x_2$, for $0<t_1<s<t_2$?
My try: First i tried to ...
1
vote
1answer
168 views
How do you show this is a martingale?
How do you show the following process is a martingale? My notes say it is a martingale by I can't work it out.
$$
E[e^{\sigma B(t) - \frac{\sigma ^2 t}{2}} | \mathscr{F}(s)]
$$
I tried to multiply ...
0
votes
2answers
192 views
Conditional Expectation of integral of Wiener process
Let $W_t$ be a standard Wiener process. How can we calculate: $$\mathbb{E}\left[\int_0^t|W_r|^2\text{d}r \ |\ \mathcal{F}_s\right]$$
where $(\mathcal{F}_s)_{s\geq0}$ is the natural filtration?
1
vote
1answer
128 views
Expectation of Stopping Time w.r.t a Brownian Motion
How do you take the expectation of a stopping time with respect to a Brownian motion? The specific question is:
$$
\tau = \inf\{ t \ge 0: B(t) \in \{-a, b\}\}
$$
I understand the optional stopping ...
2
votes
1answer
69 views
Applying Ergodic Theorem on fractional Brownian motion
For a fractional Brownian motion $B_H$ consider the sequence for $p>0$
$$Y_{n,p}={1\over n}\sum\limits_{i=1}^n \left|B_H(i)-B_H(i-1)\right|^p.$$
By the Ergodic Theorem it is ...
3
votes
2answers
258 views
Show that this process is a martingale
Let $B_t$ be a Brownian motion and $M_t=\max_{0\leq s\leq t}B_s$. Show that: $$(M_t-B_t)^4-6t(M_t-B_t)^2+3t^2$$
is a martingale for $t\geq0$.
