0
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4 views

Discrete and Continuous Time Markov Properties

$\newcommand{\indep}{\perp\!\!\!\perp}$ In discrete time, the Markov property is $$P[X_{n+1}\in A\mid X_n=s_n,X_{n-1}=s_{n-1}\dots ]=P[X_{n+1}\in A\mid X_n=s_n]$$ On the other hand, the "general ...
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1answer
15 views

Q-matrix vs. P-matrix description of a Markov chain

Consider a continuous time Markov chain $(X_t)_{t \geq 0}$ on some state space $S$ with transition matrix (P-matrix) $p_t(x,y)$, the probability density of jumping from $x$ to $y$ in time $t$. The ...
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1answer
35 views

Property of Wiener process sample path

What is a mean of time, when the trajectory of wiener process $W_t$ is over the line $y=t$? We need to find $\mathbb{E}\tau$, where $\tau=\sum\limits_{a,b:\forall t\in(a,b) ; ...
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0answers
6 views

Branching Brownian Motion and KPP equation

I have troubles understanding the proof of the connection between BBM and KPP equation. I mean the proof of the next lemma from the lecture notes of Anton Bovier about BBM, link. This is almost whole ...
0
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0answers
13 views

A question in a textbook about Blumenthal 0-1 law for a general Markov process

This question came up as a result of reading this question . Here is the Blumenthal 0-1 law in the book Stochastic Processes by Richard F. Bass. Proposition 20.8 Let $(X_t , \Bbb{P}^x)$ be a ...
1
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1answer
56 views

random walk with sticky barriers

Consider a random walk on the line 1,...,d. You start at point 1. At each step you flip a coin: heads means go left, tails means go right. If you're at 1 and get a heads, just stay where you are ...
0
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1answer
14 views

Injections of Markov Processes

In office hours, a professor mentioned that an injective transformation of a Markov process remains Markov. Intuitively, this makes sense to me as you can "recover" the original Markov process from ...
0
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1answer
17 views

Markov Processes: How to show $\int P(X_t\in B\mid X_0)dPX_0^{-1}=P(X_t\in B)$?

Let $\left\{X_t \right\}_{t\in T}$ be a time homogeneous Markov process with state space $S$. How do I formally demonstrate$$P(X_t\in B)=\int_S P(X_t\in B\mid X_0)dPX_0^{-1}$$(here $PX_0^{-1}$ is the ...
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0answers
12 views

How to use symmetry of transition rate matrix in a continuous-time Markov chain?

This is part of a bigger question, so I have to change the question a bit to focus on the point. We have a continuous- time Markov chain with the following transition rate matrix: $$Q= \begin{pmatrix} ...
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0answers
47 views

Strong Markov property for Poisson point process

The question is thoroughly contained in the title. I just say that I would only like to find a reference for this question. I have searched in some books, to no avail. Just to avoid misunderstanding, ...
0
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1answer
18 views

A problem on Markov process

Suppose, $\Pi_{\theta}$ be the transition probability function of a Markov chain. For any function $f$ define $$\Pi_{\theta}f_{\theta}(x) = \int f(y,\theta)\Pi_{\theta}(x,dy).$$ Is there any ...
1
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1answer
26 views

Reflection Principle interpretation

Given a standard Brownian motion $(\Omega,\mathcal{F},(\mathcal{F}_t)_t,\mathbb{P},(B_t)_t)$ (the standard filtration $(\mathcal{F}_t)_t$), we define $$\forall t\ge 0: M_t:=\max_{0\le s\le t} B_s$$ ...
0
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2answers
48 views

Prove that something is a Markov chain

Let $\xi_0, \xi_1, \xi_2, ...$be independent, identically distributed, integer valued random variables. Define $Y_n$ = max{$\xi_i: 0 \leq i \leq n$}. Show that $(Y_{n)n\geq0}$ is a Markov chain and ...
1
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0answers
21 views

Analysis of Steady State Probability for Markov Process

I have a balance equation, representing a Markov Chain, which yields $$ (K - z) \pi(Z_c = z) = (\lambda_c + (z+1)x) \pi(Z_c = z+1) $$ where K is the maximum state of the server. The term $\lambda_c$ ...
1
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0answers
19 views

When are the marginals of an extremal invariant measure also extremal invariant?

Let's suppose that $X$ is a compact metric space, and thus as is $X \times X$. If given a Markov process on $X \times X$ with marginals that are Markov processes on $X$, then we know that the ...
1
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1answer
29 views

$X_n, n> 0$ is a Markov Chain, how to interpret $Z_n = (X_n,X_{n+1}), n > 0$?

Am a newbie to Markov Chain. So, this might be incredibly naive/stupid question. If $X_n, \, n > 0$ is MC, am having difficulty imagining/interpreting process $Z_n = (X_n,X_{n+1}), n > 0$. I ...
0
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0answers
48 views

Stationary distribution of a “birth-death model” that does not have Markov property

A typical birth-death process is defined such as the probability of going from any state $j$ to any state $i$ is given by: $$ p_{ij}= \begin{cases} b_i & \text {if $j = i+1$} \\ ...
2
votes
1answer
63 views

continuous time Markov chain, something the book does not explain

I have a problem with something in my book, under the chapter of continuous time Markov chains. I will post a link to what the book does. They do something which they seem to take for granted, but I ...
0
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1answer
43 views

Representation of Markov process adapted to given Filtration

Let $X$ be continuous Markov process adapted to a filtration generated by Brownian motion $B$. Does there exist a function $f$ such that $X_t = f(t,B_t)$? My guess is that it should have such ...
0
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0answers
43 views

Continuous time Markov chains, how would this definition be expanded from time-homogeneous to time-inhomogeneous.

Below I have a picture of how we can view a continuous time Markov chain that is time-homogeneous. Now, I am wondering what happens when we have a inhomogeneous continuous Markov chain. I have ...
0
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1answer
50 views

Markov processes and semimartingales

Semimartingales and Markov processes are two fundamental families in probability theory. There are many specific processes that belongs to the intersection of those two families, e.g. Levy processes. ...
7
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2answers
232 views

What is the importance of the infinitesimal generator of Brownian motion?

I have read that the infinitesimal generator of Brownian motion is $\frac{1}{2}\small\triangle$. Unfortunately, I have no background in semigroup theory, and the expositions of semigroup theory I have ...
1
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2answers
93 views

Expected number of steps between states in a Markov Chain

Suppose I am given a state space $S=\{0,1,2,3\}$ with transition probability matrix $\mathbf{P}= \begin{bmatrix} \frac{2}{3} & \frac{1}{3} & 0 & 0 \\[0.3em] \frac{2}{3} ...
0
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1answer
73 views

Exercise on Markov chain

Prove, or give an explicit counterexample to refute, the following assertion: if $\{X_n\}$ is a Markov chain, then $\{X_n^2\}$ is also a Markov chain. It's easy to show that ...
3
votes
1answer
64 views

Question about Markov chain

We know that if $\{X_n\}$ is a Markov chain, then $X_{n+1}$ is independent with the past states $X_0,\ldots,X_{n-1}$ given current state $X_n$, that is ...
2
votes
1answer
103 views

Markov chain problem with finite states

Consider any Markov chain on a state space with exactly $r$ states. We want to find the largest $N>0$ such that there exists states $i,j$ for every $n < N$ we have $p_{ij}^{(N)} > 0$ and ...
1
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0answers
37 views

Markov Chain depicting unruly customer behavior

A store has 2 bins of balls. 1 bin is red, and contains 3 red balls. The other bin is gray and contains 2 gray balls. Every minute, on the minute, exactly one customer comes by the bins, picks up ...
1
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1answer
64 views

Library chain stationary distribution

This is an exercise 1.47 from Richard Durrett's Essentials of Stochastic Processes p.85 (doi: 10.1007/978-1-4614-3615-7_1 or Google Books). On each request the ith of the $n$ possible books is the ...
4
votes
1answer
77 views

random walk in a certain environment

Consider the following random walk in one dimension, starting from $r(0)=0$. $$ r(i+1) = r(i) + \xi, $$ where $\xi(i, r(i))$ is an increment with distribution $P(\xi=1) = \frac{c^{r(i)}}{i-r(i)+1}$ ...
2
votes
0answers
48 views

A theorem in the paper “Noncommuting Random Products” by Furstenberg

I have a question concerning the proof of theorem 2.5 at page 395 of the paper Noncommuting Random Products, by H. Furstenberg, Trans. Amer. Math. Soc., 1963. The statement is as follows: Let $\mu$ ...
5
votes
1answer
113 views

Log-likelihood function

I'm not sure if this could be asked here, or in math overflow... In the following paper Cho, Jin Seo, and Halbert White. "Testing for regime switching." Econometrica 75.6 (2007): 1671-1720. doi: ...
2
votes
1answer
117 views

Equivalent defining Markov property

Consider the stochastic process $(X_t)_{t \in \mathbb{R}}$ and show the equivalence of the following two Markov properties: (a) $P(X_t \in A \mid X_u, u \leq s) = P(X_t \in A\mid X_s) \qquad ...
1
vote
1answer
51 views

Strong Markov property given transition functions

Suppose we are given family of transition functions satisfying Chapman-Kolmogorov equation, what conditions will ensure that there exists a continuous or cadlag Markov process with given transition ...
0
votes
1answer
50 views

Stationary distribution for a Markov chain which is not irreducible

I have a Markov chain with $K$ states $S$: {$s_1,s_2,...,s_K$}. $s_1$ is reachable from any state in $S$; however not all the states can be reached from $s_1$. What does the stationary distribution ...
0
votes
0answers
52 views

Continuous-Time Markov Chain conditioned on not visiting a part of the state space

Let $(X_t)_{t\geq 0}$ be a homogeneous continuous-time Markov chain with state space $\Psi =\{1,\dots,N\}$. Consider $S=\{1,\dots,M\} \subset \Psi$. Define $T = \inf \{ t \geq 0 : (X_t \not \in S ...
2
votes
1answer
82 views

Markov chain and hitting times

I have a Problem about hitting times. That's the following: Let $A\subset E$ and the first passage time $T_A$ and the hitting time $H_A$. Define: $T_A =\inf\{n\geq 0;X_n \in A\}$ and $H_A ...
2
votes
1answer
64 views

transition kernel

I've got some trouble with transition kernels. We look at Markov process with statspace $(S,\mathcal{S})$ and initial distribution $\mu^0$. We have a transition kernel $P:S\times ...
0
votes
0answers
79 views

Markov Chains Proof using Statistics

Source: This came from "Introduction to probability" by Charles Miller Grinstead, and James Aurie Snell. It was located on page 407 and is Theorem 11.1 in the section 11.1 Introduction. Theorem: The ...
1
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1answer
63 views

Transition function is a Markov semigroup?

How does the transition function in a Markov process become a Markov semigroup in time homogeneous Markov processes? Thanks a lot.
1
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2answers
146 views

Question on the proof of the simple Markov property of a Brownian motion

Today we proofed the (simple) Markov property for the Brownian motion. But I really don't get a crucial step in the proof. The theorem states in particular that for $s\geq0$ fixed, the process ...
1
vote
1answer
91 views

How is a simple birth process is time-homogeneous?

Why is it that a simple birth process is time-homogeneous? The incidence of a birth in a small time interval depends on the population size at the time of start of the interval. Doesn't this ...
2
votes
0answers
66 views

Interpretation of potential kernels for Markov processes

One can associate a strongly continuous contraction semi group (SCCSG) to a Markov process with state space $S$ through its transition function, say $P_t$. Now one can interpret $P_t$ as a linear ...
1
vote
4answers
171 views

Why do these expressions tend to zero?

In a chapter on Markov chains, it is claimed that $\binom {2n}{n} p^n(1-p)^n$ (where $p$ is a probability) tends to $0$ as $n$ tends to $\infty$. But why is this so? It is also claimed that for an ...
0
votes
0answers
67 views

Formal Theory Regarding M/M/s Queue

I have some difficulties with formally deducing the Q-matrix or infinitesimal generator for M/M/s Queues. Although I undestrand the intuitive idea I would like to know the real formal definition of ...
-1
votes
1answer
319 views

What is the Expectations of all 3 ants meeting at same point?

Say we have 3 ants in three corner's of triangle. What is the expectations that all 3 ants meeting together given that the ant moves in any direction. So by just seeing it I figured out that in 2 ...
0
votes
1answer
26 views

Is it possible to reverse probabilistic automaton?

Is it possible to reverse probabilistic automaton (PA), i.e. calculate the probability of previous state given current state? Will reversed automaton be a PA (Markov?), i.e. will next probability ...
2
votes
1answer
49 views

Markov property of a random process (a solution of piece-wise deterministic equations)

Consider a piece-wise deterministic (Markov!) process
3
votes
1answer
71 views

A equivalent definition of the Feller Process.

I saw this on Liggett's Book (P.95). Let $S=% %TCIMACRO{\U{2115} }% %BeginExpansion \mathbb{N} %EndExpansion ,$ and suppose $\left( X_{t}\right) _{t\geq 0}$ is a continuous-time Markov process with ...
0
votes
1answer
56 views

On discrete-time stochastic attractivity

Let $m$ be a probability measure on $Y \subseteq \mathbb{R}^p$, so that $m(Y)=1$. Consider a function $f: \mathbb{R}^n \times Y \rightarrow \mathbb{R}^n$, continuous on the first arguments, ...
3
votes
2answers
449 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 ...