Tagged Questions
0
votes
1answer
19 views
Examples of convergence of random variables
First, let's recall the definitions of 4 different types of convergence:almost surely, in $r$th mean, in probability and in distribution:
$X_n\xrightarrow{a.s.}X$ if $\{\omega \in ...
0
votes
1answer
31 views
Continuous Non negative martingale converging to 0
Is there any (non trivial) continuous non negative martingale which converges to 0?
0
votes
1answer
23 views
Generalization of Doob Dynkin for Stochastic processes
Let $\{X_t\}_{t\geq 0}$ be continuous time stochastic process and $\{\mathcal{F}_t^X\}_{t \geq 0}$ be the filtration generated by it. If the process $Y$ is $\{\mathcal{F}_t^X\}_{t \geq 0}$ adapted, is ...
2
votes
0answers
25 views
lower bound of expectation of stochastic differential equation
I'm looking for a lower bound on the expected value of a smooth, non-negative, increasing function $\mathbb{E}f(X_t)$, $f(0)=0$ of the solution to a stochastic differential equation $X_t = x + ...
1
vote
1answer
42 views
finding the probability density function of $ dY_t = - Y_t X_t dW_t$
Could someone point me to where I can learn how to derive the stationary distribution for the martingale $Y_t$ which itself has stochastic volatility drive by $X_t$:
\begin{align}
dY_t &= - Y_t\ ...
1
vote
1answer
33 views
The weighted distribution function for combination of two variables
For example, we have two random variables $a$ and $b$. And they have cumulative distribution function $F(x)$ and $H(x)$. We have number $0 < p < 1$.
Suppose, some machine get this random ...
1
vote
0answers
21 views
Girsanov kernel moments
Let $Z_t=e^{\int_0^tq_tdB_t-\frac{1}{2}\int_0^tq^2_tdt}$, where $(q_t)_{t\geq0}$ is a predictable process, and $(B_t)_{t\geq0}$ a $\mathbb{P}$-Brownian motion. In particular, Novikov's condition ...
0
votes
1answer
35 views
Continuous time Stochastic Process stopping time measurability
Let $\{X_t,\mathcal{F}_t;0\leq t < \infty\}$ be continuous time stochastic processes and $T$ be $\{\mathcal{F}_t\}_{0\leq t < \infty}$ stopping time. How to prove $X_T$ is $\mathcal{F}_T$ ...
0
votes
0answers
13 views
fGn asymptotic claim correlation
Let $(X_{i})$ be the fractional Gaussian noise for $H\in(0,1)$.
Since it is stationary $\mathbb{E}(X_{i}X_{j})$ only depends on $|j-i|$.
How can I prove for $\rho(|j-i|)=\mathbb{E}(X_{i}X_{j})$ that ...
1
vote
1answer
36 views
Moment generating function of two non-independent Brownian increments
I am writing to ask if it is possible to get closed-form solution to the expression to the following expression:
$\mathbb{E}[e^{\sigma^2(W_t-W_u)(W_s-W_u)}]$ where $W$ is a standard Brownian motion, ...
-1
votes
0answers
20 views
Random process x(t) =C and C is uniform over [-2,3]
I need reassurance that if I do a a few sample realizations of this random process they are all going to look the same. They are going to be an horizontal line with x(t) constant equal to 1/5.
I see ...
-2
votes
0answers
31 views
Transforming a Joint PDF [duplicate]
I have a pdf $f(X,Y)=(\frac{1}{4})^2e^{−\frac{(|x|+|y|)}{2}}$. My goal is to find the joint PDF $f(W,Z)$ taking in consideration this $W=XY$ and $Z=Y/X$.
I know I can not use Jacobian because is a ...
0
votes
0answers
13 views
Submartingale bounds
Let $X_1,X_2,\ldots$ be a submartingale with respect to the filtration generated by it. Is it possible to get any bounds for the probability $\mathbb{P}(X_2 < 0\mid X_1 >0)$ ?
1
vote
1answer
76 views
A Boundary crossing result for discrete brownian bridge
Let $S_n$ be a random walk with gaussian increments with $S_0=0$, i.e. $S_n-S_{n-1}\sim N(0,1), n\geq 1$. Fix $a>0,b\in \mathbb{R}$ and $c<a+bn$. Define the new process
$$
...
2
votes
1answer
62 views
Sum of stationary process
Suppose you have two stationary process $A_{t}$ and $B_{t}$. Suppose $Z_{t} = A_{t} + B_{t}$. Show that $Z_{t}$ is stationary. I am unsure how to solve this without knowing if the processes are ...
0
votes
1answer
38 views
Product of stationary stochastic process
Suppose $z_{t} = x_{t}y_{t}$ where $x_{t}$ and $y_{t}$ are 0 mean, independent stationary stochastic process. What is the autocovariance function of $z_{t}$? Show that the spectral density can be ...
1
vote
1answer
36 views
Finding expectation of stochastic process
Suppose $\sigma_{t}^{2} = w + \alpha_{1}y_{t-1}^{2} + \beta_{1}\sigma^{2}_{t-1}$ where $\alpha_{1} + \beta_{1} = 1$ and $y_{t} = \sigma_{t}e_{t}$ and $e_{t}$ is $ N(0,1)$. How do you show that
...
1
vote
1answer
30 views
Inequality related to Doob's martingales
I have the following question on Doob's martingales.
Let $A$ be an integrable $\mathcal F$-measurable random variable on the
stochastic basis $(\Omega, \mathcal F, \mathcal F_t, \mathbb P)$. ...
0
votes
1answer
36 views
Integral: Is there a closed form?
I wonder whether there is a closed form or way to compute explicitly:
$$\int_0^t e^{\alpha s} dB_s$$
where $\alpha$ is just a real number and the integral is in the Itô sense.
Thank you very much!
1
vote
1answer
58 views
Using a Brownian martingale to compute the second moment of a hitting time
Prove $ W_t=B_t^4 -6B_t^2t+3t^2$ is a martingale, and compute $E(T^2)$ where $T=\inf(t\ge0,B_t=-a, B_t=b)$ if $a=b$.
Ok, if $0\lt t\lt s$, $W_t$ is a martingale if $E(W_s|[B_r]_{r\le t})=W_t$
So ...
0
votes
1answer
23 views
Basic brownian motion computation
Let $B_t$ denote a standard 1-d Brownian Motion. Find $P(B_2 \gt 2)$.
My sol.
$B_2 ~ N(0,2)$ so $P(B_2 \gt 2)=1-P(B_2\le 2)=1-\frac{\int_0^2e^{-\frac{x^2}{4}}}{\sqrt{4\pi}}$, but where do i go from ...
1
vote
0answers
29 views
Cylindrical sigma algebra answers countable questions only.
I got a missing link in some in the following (standard) textbook question:
Show that the cylindrical sigma algebra $\mathcal{F}_T$ on $\mathbb{R}^T$ (equals $\bigotimes_{t\in ...
4
votes
2answers
78 views
Prove that a random walk on $\mathbb{Z}_+\cup \{0\}$ is transient
Prove that a random walk on $\mathbb{Z}_+ \cup \{0\}$ is transient with $p_{i,i+1}=\frac{i^2+2i+1}{2i^2+2i+1}$ and $p_{i,i-1}=\frac{i^2}{2i^2+2i+1}$.
So since this Markov chain has only a single ...
2
votes
1answer
57 views
A bound for the probability that a Brownian motion stays in an interval
Suppose I have a Brownian motion $X_t$ with $X_0=0$. Let $T$ be the first exit time of the interval $[-1,1]$.
I'm trying to get a "quick" lower bound for the probability that $T$ is very large which ...
0
votes
0answers
27 views
Computing spectral density of process
Suppose you have a stochastic process $Y_{t} = \frac{1}{2}(X_{t-1} + X_{t} + X_{t+1})$. $X_{t} = 0.4X_{t-1} + \omega_{t}$. How would you compute the spectral density of the process? I know that ...
1
vote
1answer
42 views
Is geometric Brownian motion stationary?
I was just wondering if the solution to
$$dX(t) = \mu X(t) dt + \sigma X(t) dB(t)$$
gives a stationary process for any $\mu,\sigma$ and what the distribution would be.
Thanks a lot!
1
vote
1answer
30 views
Normal probability and Brownian motion
Let $X_t$ be a Brownian motion with parameter $\sigma$. Find the probability in terms of $$\Phi(x)= \frac{1}{\sqrt{2 \pi}} \int_{- \infty}^x e^{- \frac{ \alpha ^2}{2}}d\alpha$$
How would I do this for ...
3
votes
0answers
53 views
Prove the 2 definitions of the periodicity of Markov Chain are equivalent.
In many textbooks, there are basically 2 ways of defining the periodicity of Markov Chain. One is by partitioning the graph in to subgraph such that transition in one group of state leads to the other ...
0
votes
0answers
8 views
Measuring time with a clock that monitors decay events occurring with a known mean time (though sampling from an unknown probability distribution)
Imagine I have some hypothetical particle that decays over time, where $\mu$ is the mean decay time, and where the probability of each decay event is governed by some unknown probability distribution. ...
0
votes
0answers
23 views
Densities of r.v in stochastic analysis
I have several exercises to solve and there are two which I somehow do not manage to solve...
We consider $W=\{W_t:t\geq0\}$ a standard B.M. issued from zero, for $a\in \mathbb{R}$, ...
0
votes
0answers
21 views
How to calculate this conditional expectation in a bandit problem?
Supposing we have different $I$ products and we want to determine the prices that maximizes the total reward (equivalently minimizes the regret) after $T$ times (bandit theory).
I'm working using a ...
0
votes
0answers
22 views
Is this probabilistic principle for stochastic processes known?
This is a cross-post for mathoverflow #125058: http://mathoverflow.net/questions/125058/is-this-probabilistic-principle-for-stochastic-processes-known. I got a single answer, which I couldn't ...
2
votes
1answer
49 views
A problem related with sum of uniform variables
This problem appeared in my mid-term exam.
Let $U_n$, $n\ge1$, be i.i.d. random variables which are uniform in $(0, 1)$.
Given a constant $t>0$, let $N$ denote the value of $n$ such that
...
0
votes
1answer
39 views
Poisson process expected value
Let {${N(t), t \geq0}$} be a Poisson process with rate $\lambda$. Let $S_n$ denote the time of the $nth$ event. What is $E[N(4)-N(2)|N(1)=3]$?
(Note: $E[X]$ is the expected value or mean).
I know ...
0
votes
1answer
53 views
Proving a Probability Generating Function satisfies a partial differential Equation
We have N animals grazing in a field. The animals graze independently, and periods of grazing and resting alternate for the animals. If an animal is resting at time t, the probability it begins ...
3
votes
2answers
75 views
Probability of Extinction in a simple Birth and Death Process
We are asked to show that the probability of extinction $\zeta=\lim_{t\to \infty} P\left(X(t)=0\right)$ given by:
$$\zeta=\begin{cases}1&\text{if }\lambda\le \mu,\\
\left(\frac \mu\lambda ...
-2
votes
1answer
43 views
Please help me with the proof on conditional variance
Let $X$ be a square integrable random variable on $(\Omega,\mathcal{F},P)$. Let $\mathcal{G}$ be a sub-$\sigma$-algebra of $\mathcal{F}$. Define the conditional variance of $X$ given $\mathcal{G}$ by ...
1
vote
0answers
38 views
Rate of increase of maximum process of Brownian Motion
Suppose $M_t=\sup_{0\leq s\leq t}\{B_s\}$, where $\{B_t\}_0^{\infty}$ is a standard Brownian Motion. I would like to know if it is true that $M_t e^{-t}$ converges to 0 almost surely?
Thanks!
3
votes
1answer
115 views
A Coupled Random Walk on the xy-Plane
Consider a point on the $xy$-plane whose position is updated in iterations. In each iteration, the point undergoes, with equal probability, either an $A$- or a $B$-update, defined as follows:
...
1
vote
1answer
36 views
Expected time, exponential distribution
Suppose that you arrive at a single-teller bank to find five other customers in the bank, one being served and the other four waiting in line. You join the end of the line. If the service times are ...
0
votes
0answers
28 views
Exponential distribution probability
$T$ is an exponentially distributed random variable.
$T$= the time required to repair a machine; with mean $\frac{1}{2}$ hours.
For the first part of the question, I am asked to find the ...
3
votes
0answers
60 views
Expected value with a kronecker product and Gaussian distributional assumption
What is the expected value, $ \mathbb{E}\left[ I \otimes \left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right)\right]$ where $Z \sim N(0, \sigma^2I) $ is a random variable? The kronecker product ...
1
vote
1answer
66 views
Martingale inequality
Let $f: \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}$ be a deterministic function, as nice as you want, $W$ a Brownian motion and define
$$
Y^r_t := \int_0^t f(r,s) dW_s
$$
For each fixed $r$, ...
2
votes
1answer
94 views
Condition Expectation of Difference between Two Poisson processes
$P_t$ and $Q_t$ are poisson processes with rates $a$ and $b$.
How do I calculate $E[(P_t-Q_t)]^2|Q_t=m-P_t]$?
1
vote
1answer
104 views
Conditional variance of arrival times
Given a poisson process $P_t$ with rate $r$, with arrival times $S_n$
How do I calculate the Variance of $S_2-S_1|P_t=2$?
0
votes
1answer
97 views
Conditional CDF of Poisson process
$X_t$ and $Y_t$ are poisson processes with rates $a$ and $b$ (independent processes)
$n = 1,2,3...$
Find the conditional CDF $F_X{}_t{}_|{}_X{}_t{}_+{}_Y{}_t{}_={}_n(x)$
I get an answer of ...
1
vote
1answer
74 views
Probability of obtaining one infinitely often in a sequence of Bernoulli trials
Consider a sequence of Bernoulli trials $X_{1}, X_{2}, X_{3}, \dots$, where $X_{n}=1$ or $0$. Assume:
\begin{equation}
P\{X_{n}=1 \mid X_{1}, X_{2}, X_{3}, \dots, X_{n-1}\}\geq \alpha>0, \; ...
1
vote
2answers
101 views
Probability of visiting state $s_1$ of a Markov chain more than $N$ times in $L$ steps.
Assume we have a two-state Markov chain, with $s_1$ and $s_0$ denoting the two states. The initial state of the Markov chain is either $s_1$ or $s_0$ with probability $p_1$ or $p_0$, respectively. The ...
0
votes
1answer
60 views
Pure Death Process Question
I am reviewing for an exam and ran across the following problem. I do not understand pure birth and death processes very well and I was hoping someone could walk me through the following problem and ...
2
votes
1answer
53 views
Probability - Poisson Process and Conditional Expectation
Suppose that $N=\{N(t); t \geq 0\}$ is a Poisson Process, with the intensity $\lambda$.
a. Find the probability that the value $N(t)$ is odd (at a given time $t$)
b. Find the conditional expectation ...
