A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.
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17 views
Change of probability measure and a continuous-time Markov chain
Let $(\Omega,\mathcal{F},\mathbb{P},\mathbb{F})$ be a complete filtered probability space, with $W$ a Wiener process and $\alpha$ a continuous-time Markov chain (taking values in $\{1,...,M\}$). We ...
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0answers
15 views
Modified M/M/1/2 with 2 possible arrival rates and M/M/1/5 queue
I've been stuck on this question for hours, and could use some help :)
"An M/M/1/2 queue has service rate $\mu$ and arrival rate of either $\lambda_1$ or $\lambda_2$. The rate can change only when ...
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0answers
16 views
How to calculate auto-correlation of a bpsk modulated signal or how to calculate expectation value of complex exponential function [migrated]
How to calculate auto-correlation of a bpsk modulated signal or how to calculate expectation value of complex exponential function manually not by using matlab or any other software? for example,if ...
1
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0answers
21 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
70 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
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1answer
33 views
Showing a conditional Poisson Process does not have independent increments
Suppose $N\left(\cdot\right)$ is a Poisson Process with rate $1$ and $Z$ is a positive non-constant random variable, define $N_{Z}\left(t\right)=N\left(Zt\right)$. I know that conditional on $Z$ this ...
1
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1answer
32 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 ...
2
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1answer
32 views
Weak convergence and generating function
Let $X_n$ be a sequence of $\mathbb N_0$ valued random variables and denote by $g_{X_n}$ their generating function, i.e. $g_{X_n}(s) = \mathbb E[s^{X_n}] = \sum_{k=0}^{\infty} s^k \mathbb P(X_n=k)$.
...
0
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0answers
18 views
Variability in estimations over a non-ergodic/non-regular Markov process
Imagine we have a non-ergodic/non-regular Markov Process with with $n$ states.
Among these $n$ states, there are $k$ absorbing states.
For each of the $n-k$ non-absorbing states, it is not possible ...
4
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1answer
29 views
Question regarding Poisson process
Let $N\left(\cdot\right)$
be a Poisson process with rate $1$ and let $\Lambda\left(t\right)$
be a non-decreasing right-continuous function. Define ...
5
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0answers
42 views
+50
diagonalization procedure for stochastic processes
This is a question about a proof I saw in a script about stochastic processes. First I state a theorem which is needed in the proof. After that there are two questions, which are highlighted. Between ...
2
votes
0answers
24 views
Supermartingale result from Meyer Dellacherie
Suppose I have a stochastic process $(X_r)$ with $r\in\mathbb{Q}\cap [0,T]$. Furthermore I know that there is a sequence of stochastic processes $Y^n$ (each a supermartingale for every $n$) such that ...
2
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1answer
28 views
Martingale equality
The question is to prove
$$P\{\sup_{t\geq 0}M_{t}>x\mid \mathcal{F}_{0}\}=\min\left\{1,\frac{M_{0}}{x}\right\},$$
where $M$ is a positive continuous martingale which converges to 0 almost surely ...
1
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0answers
14 views
Analytic random function
Let $t\in[0,1]\mapsto X_t\in[0,1]$ an analytic random function.
I'd like to say that the deterministic function $t\mapsto \mathbb{E}[X_t]$ is still analytic.
What are the minimal conditions needed? ...
7
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2answers
49 views
Two martingales whose distributions agree for each time have the same overall distribution
Let $\{X_n\}$ and $\{Y_n\}$ be two martingales. Suppose that for each fixed $n \in \mathbb Z_+$, $X_n$ and $Y_n$ have the same distribution. Must it hold that the random sequences $\{X_n\}$ and ...
0
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1answer
38 views
About equivalent characterization of ergodicity
Can anyone give me some hint on the following problem? Many thanks!
Given a probability space $(X, \Sigma, \mathbb{P})$ and a $\mathbb{P}$-preserving map $\tau: X\to X$, show that the following three ...
1
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1answer
26 views
Inequality- Absolute Value general powers
Iam trying to understand the following inequality:$p>0$
Let $$T_{m}:=\sum_{i=1}^{m}\left(\left|\int_{\frac{i}{n}}^{\frac{i-1}{n}}g(s)dW_{s}\right|^{p}-\left|g ...
0
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0answers
25 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 ...
0
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0answers
15 views
Find the distribution of the increments from Langevin equation?
Given a Langevin eq. of a stochastic process:
X[I+1]=X[I]-F(X[I])+W[I]
- where F(X[I]) is a position dependent force, and W[I] is the Wiener process term (i.e. Gaussian, white-noise).
How do I ...
2
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0answers
28 views
Question on M/M/2 queue variation
I have the following question:
Two workers handle three machines(i.e. we can at most repair two machines at a time). The time until the machine breaks down is exponential distributed with expectation ...
0
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0answers
37 views
Question about Infinite Markov chains
Do 2 Markov chains $\left\{X_n\right\}^\inf_{n=0} $ and $\left\{Y_n\right\}^\inf_{n=0} $ with all of these properties exist so that the probability for infinite n values to maintain $X_n=Y_n$ is 0? ...
2
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1answer
54 views
$\sigma$-algebras and independent stochastic processes
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a complete probability space. We consider a Wiener process $W$ with respect to his standard filtration $(\mathcal{F}_t^W)_{t \geq 0}$ and a process $X$ with ...
2
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0answers
68 views
Pure birth process
Let $\lambda_n$ denote the arrival rate for a pure birth process of size $n$. Let $P_n(t)$ denote the probability of population size $n$ at time $t$.
A stochastic process is dishonest if ...
1
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1answer
36 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)$$
gives a stationary process for any $\mu,\sigma$ and what the distribution would be.
Thanks a lot!
1
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1answer
26 views
Normal probability and Brownian motion
Let $X_t$ be a Brownian motion with parameter $\sigma$. Find the probability in terms of $$\Phi(x)= \frac{1}{\sqrt{2 \pi}} \int_{- \infty}^x e^{- \frac{ \alpha ^2}{2}}d\alpha$$
How would I do this for ...
3
votes
0answers
41 views
Prove the 2 definitions of the periodicity of Markov Chain are equivalent.
In many textbooks, there are basically 2 ways of defining the periodicity of Markov Chain. One is by partitioning the graph in to subgraph such that transition in one group of state leads to the other ...
0
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0answers
31 views
Drift equation / Girsanov's Theorem
Define $$\frac{dS_t}{S_t} := \mu \, dt + \sigma^B \, dB_t +\sigma^W \, dW_t,$$ $$dS_t^{(0)} := S_t^{(0)}r \, dt,$$ where the constants $\mu, \sigma^B,\sigma^W$ and $r$ all positive and $\mathcal{F}_t$ ...
0
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0answers
17 views
SDE(s) satisfied by Radon Nikodym derivatives of martingale measures?
Given:
Money Market Account: $dR_{t}=R_{t}r_{t}dt, R_{0}>0$
Risky Asset: $dS_{t}=S_{t}(\mu_{t}dt+\sigma_{t}dB_{t}), S_{0}>0$,
where $r, \mu,$ and $\sigma$ are positive processes and $B$ is a ...
0
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0answers
19 views
Supermartingale Lemma + related problems
Given the following Lemma: Let $A_{t}=\int_{0}^{t}a_{s}dB_{s}$ where $a$ is an adapted process satisfying $\mathbb{P}\Big(\int_{0}^{T}a^{2}_{u}du < \infty\Big) = 1$ and $B$ is a standard Brownian ...
0
votes
1answer
25 views
Finite expectation
Let $X_1,X_2,...$ be i.i.d with mean $\mu$. Let $T$ be a stopping time with respect to $X_,X_2,...$ with $E(T)<\infty$. Show that $E\left(\sum_{n=1}^\infty |X_n|I\{T\ge n\}\right)<\infty$.
My ...
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votes
1answer
20 views
Linear prediction of weakly stationary stochastic process
Let $X\{(n)\}_{n\in \mathbb Z}$ be a weakly stationary stochastic process.
Given the information up to time $n$, $\{X_k\vert k\leq n\}$, in what way does the optimal linear predition change when I ...
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0answers
15 views
Is open-ball-weak convergence of borel-measurable random elements the same as borel-weak-convergence?
The definitions are from David Pollard, Convergence of Stochastic Processes, IV.1.1 p.65
Let $(\Omega,\mathcal{A},P)$ be any probability space.
Let $(S,\mathcal{S})$ be a metric space with any ...
1
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3answers
24 views
Scaling and time inversion for Brownian motion basically the same?
Let $B(t)$ be a Brownian motion. For $a>0$, we have the scaling relation
$$\hat{B}(t)=aB(t/a^2) \sim B(t)$$
and $\hat{B}(t)$ is also a Brownian motion.
The time inversion formula states that
...
0
votes
0answers
19 views
On discrete-time stochastic attractivity of linear systems
Let $m$ be a probability measure on $Y \subseteq \mathbb{R}^p$, so that $m(Y)=1$.
Consider a continuous function $f: \mathbb{R}^n \rightarrow \mathbb{R}^p$. Assume that $f(0) = 0$, and that there ...
0
votes
0answers
8 views
Measuring time with a clock that monitors decay events occurring with a known mean time (though sampling from an unknown probability distribution)
Imagine I have some hypothetical particle that decays over time, where $\mu$ is the mean decay time, and where the probability of each decay event is governed by some unknown probability distribution. ...
2
votes
1answer
47 views
Reality check: $\mathbb E \{ B_s B_t ^2\}=0 $
I desire to calculate $\mathbb E \{ B_s B_t ^2\} $, where $B$ is a standard brownian motion starting from zero. I want to be sure I am not making any mistake on both reasoning and result, even if I ...
1
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1answer
19 views
Why is there “markov property” in proving the renewal equation for a renewal process?
When proving the renewal equation for a renewal process in Wikipedia
The renewal function satisfies
$$
m(t) = F_S(t) + \int_0^t m(t-s) f_S(s)\, ds
$$
where $F_S$ is the cumulative ...
0
votes
1answer
16 views
Differences between a Markov jump process and a continuous-time discrete-state Markov process?
What are the difference and relation between a Markov jump process and a continuous-time discrete-state Markov process?
By "a continuous-time discrete-state Markov process", I understand it same as a ...
0
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0answers
22 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}$, ...
2
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1answer
26 views
Two different ways of constructing a continuous time Markov chain from discrete time one
Consider a homogeneous continuous time Markov chain (CTMC) with Markov transition function $p(t)$ and infintesimal generator $G$.
Its embeded discrete time Markov chain (DTMC) has its transition
...
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0answers
13 views
Solving for normal modes for a birth-death process with absorbing boundary conditions
Suppose I have the following birth-death process:
$
\dot{p}_n=((n-3)+1)(\alpha+\beta)p_{n+1}+((n-3)-1)\beta p_{n-1}-(\alpha(n-3)+2(n-3)\beta)p_n\\
\dot{p}_3=(\alpha+\beta)p_4-\omega p_3\\
...
0
votes
2answers
21 views
A state $i$ is recurrent, if and only there exists $n \geq 1$ s.t. $p_{ii}^{(n)} =1$?
For a homogeneous discrete time Markov chain with transition matrix $p$, a state $i$ is recurrent, if and only there exists $n \geq 1$ s.t. $p_{ii}^{(n)} =1$?
I have it copied from somewhere in my ...
-1
votes
2answers
31 views
Covariance for periodic weakly stationary process
Let $X(n),n\in \mathbb N_0$ be a weakly stationary process with $X(n) = X(n+N)$ for some $N \in \mathbb N_0$.
What is the covariance function $b(k):=\operatorname{Cov}[X(n+k),X(n)]$?
0
votes
1answer
54 views
Fail of optional sampling theorem
Could anyone help me see why the optional sampling theorem ($E(M_{\tau}\mid\mathcal{F}_{\sigma})=M_{\sigma}$ a.s.) fails for certain stopping times $\sigma\leq\tau$ for the not uniformly integrable ...
0
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0answers
14 views
a probleme about transformation of homogeneous SDE
I have a question about the characteristics of homogeneouse SDE:
$$dX_t=\beta(X_t)+\alpha(X_t)dW_t\ \ (1)$$
where $W$ implies a standard Brownian motion.
To be more specific, given a general SDE:
...
0
votes
1answer
23 views
Process with independent increments: relation of increments to process value at later time
Let $X_t,t\geq 0$ be a process with independent increments, $X_{t+s}-X_t$ is independent of $X_r,r\leq t$.
Can something similar be said about a later value and and an earlier increment, for example ...
0
votes
1answer
32 views
Exchanging limit and expectation for $L^2$ random variables
Let $X_n$ be a sequence of random variables in $L^2$, i.e. $\mathbb E[\vert X_n \vert^2]<\infty$. Since the expectation value can be interpreted as a scalar product on $L^2$, can one exchange limit ...
1
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1answer
74 views
Doob's supermartingale inequality
I'm trying to prove that For a non-negative supermartingale $M$ it holds that for all $\lambda>0$ we have
$$\lambda P\{\sup_{n}M_{n}\geq\lambda\}\leq E(M_{0})$$
My idea was to use Markov's ...
2
votes
0answers
10 views
Meyer's Theorem in Williams & Rogers
In Diffusions, Markov Processes and Martingales Volume 2 by Rogers and Williams they state the following theorem due to Meyer:
$\mathbf{Theorem }$ Le $M\in\mathcal{M}^2_0$. Then there exists a ...
0
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0answers
11 views
Concepts: time homogenous and independent increments
Can someone give me an illustrative example for a time homogenous process without independent increments and for a process that is not time homogenous, but has independent increments?
