A stochastic process satisfying the Markov property: the distribution of the future states given the value of the current state does not depend on the past states. Use this tag for general state space processes (both discrete and continuous times); use (markov-chains) for countable state space ...

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512 views

Hilbert's Barber Shop

Hilbert opens a barber shop with an infinite number of chairs and an infinite number of barbers. Customers arrive via a Poisson random process with an expected 1 person every 10 minutes. Upon arrival, ...
12
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3answers
275 views

Select a new value from last $N$ values; how long until the last $N$ are all the same?

Say first we have N distinct numbers in a line, like 1,2,3,...,N, in each round, we choose a ...
12
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2answers
426 views

Difference in probability distributions from two different kernels

I wonder if the probability kernels of Markov processes on the same state space are close enough, does it also hold for the probabilities of the event that depend only on first $n$ values of the ...
11
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1answer
440 views

Motivation of Feynman-Kac formula and its relation to Kolmogorov backward/forward equations?

Kolmogorov backward/forward equations are pdes, derived for the semigroups constructed from the Markov transition kernels. Feynman-Kac formula is also a pde corresponding to a stochastic process ...
10
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2answers
575 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 ...
8
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0answers
146 views

Uniqueness of an infinite system of linear ODEs

How to prove that $\dot{x}=ax,\space x(0)=1$ has a unique solution if $a,x$ are infinite dimensional matrices? More specifically, let $Q$ be a bounded infinitesimal generator, i.e. ...
6
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0answers
185 views

Question on Conditional expectation

Let $X_1$ and $X_2$ be two random variables on $(\Omega,\mathcal{B},P)$. Suppose there is a function $g:\mathcal{B}\times\mathbb{R}\rightarrow[0,1]$ such that for any $x$, $g(\cdot,x)$ is a ...
6
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4answers
265 views

Discover where Bob is sleeping using hidden Markov chains

Bob lives in four different houses $A, B, C$ and $D$ that are connected like the following graph shows: Bob likes to sleep in any of his houses, but they are far apart so he only sleeps in a house ...
5
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1answer
2k views

Markov process vs. markov chain vs. random process vs. stochastic process vs. collection of random variables

I'm trying to understand each of the above terms, and I'm having a lot of trouble deciphering the difference between them (note, my mathematics training isn't very strong - so please go easy on the ...
5
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1answer
110 views

How is the Chapman-Kolmogorov Equation not a fancy name for matrix multiplication?

The Chapman-Kolmogorov Equation: $$p^{m+n}(i,j)=\sum_kp^m(i,k)p^n(k,j)$$ Matrix Multiplication (with $[A]_{i,j}=a_{i,j}$ where $A$ is a linear map "" for B) $$[AB]_{i,j}=\sum_ka_{i,k}b_{k,j}$$ In ...
5
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3answers
180 views

Does an n-order Markov chain still represent a Markov process?

I am trying to understand Markov processes but am still confused by their definition. In particular, the Wikipedia page gives this example of a non-Markov process. The example is of pulling different ...
5
votes
1answer
116 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: ...
5
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1answer
2k views

Transition Kernel of a Markov Chain

Supposing $X_t$ is a Markov Process, can the transition kernel be defined by $$K_t(x,A):= P(X_{t+1} \in A | X_t = x)?$$ Assume that $X_t : \Omega \to \mathbb{R}^n$. The issue is that under the normal ...
5
votes
1answer
100 views

Conditions for birth and death process having only finitely many deaths.

Consider a birth and death process on $\mathbb{N}=\left\{0,1,2,\ldots\right\}$, given by the transition probabilities $p(n,n+1)=\lambda_n$ and $p(n,n-1)=\mu_n$ (those are the birth and death rates, ...
4
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1answer
407 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 ...
4
votes
1answer
1k views

Why does a time-homogeneous Markov process possess the Markov property?

Klenke defines (Definition 17.3, p. 346) a time-homogeneous Markov process independently, rather than as a special case of a stochastic process that possesses the Markov property (Definition 17.1, p. ...
4
votes
1answer
36 views

A die whose score cannot be as before (Markov chains)

A die is "fixed" so that each time it is rolled the score cannot be the same as the preceding score, all other scores having probability $1/5$. Given that the first score is 6, what is the probability ...
4
votes
1answer
81 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 $P(\xi=1) = \frac{c^{r(i)}}{i-r(i)+1}$ ...
4
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3answers
3k views

Why Markov matrices always have 1 as an eigenvalue

Also called stochastic matrix. Let $A=[a_{ij}]$ - matrix over $\mathbb{R}$ $0\le a_{ij} \le 1 \forall i,j$ $\sum_{j}a_{ij}=1 \forall i$ i.e the sum along each column of $A$ is 1. I ...
4
votes
1answer
96 views

Showing a process is not markov

I keep searching but I can't find any place that gives a good method of showing a process is NOT Markov. The definition I am using is that for every $s<t$ and $g$ bounded borel there is $f$ borel ...
4
votes
1answer
92 views

Fixed point of transition kernel generates martingale

Let $P^{h}, h \geqslant 0$ be a transition kernel for some homogenous Markov process $X_t$, $\mathbb{E}|X_t|<\infty$: $$ P_{X_{t+h},X_t}(A,B) = \int\limits_{A}P^h(x,B)P_{X_t}(dx) $$ where ...
4
votes
1answer
435 views

Recursively Solving a Bellman Equation

Problem Overview I am to figure out $v_\pi$ of a certain Markov state. Given Information A set of actions, $a$ containing ${up, down, left, right}$ $v_\pi(12), v_\pi(13), v_\pi(14)$ (I am given ...
4
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0answers
77 views

A Continuous-Time Markov Process Taking All Possible Values

Let $\mathbb{N}$ be the set of positive integers. For each $n \in \mathbb{N}$, let $X^{(n)}=\{ X^{(n)}(t): t \geq 0 \}$ be a Markov chain with state-space the two point set $\{0,1\}$ and $Q$-matrix ...
4
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0answers
138 views

Conditional probability and integrating out part of a random walk

Suppose that I have a random walk process defined by $\alpha_{t+1}$ ~ N$(\alpha_t, \omega^2)$. Given $\alpha_t$ and $\alpha_{t+2}$, I understand why the conditional formula for ...
4
votes
0answers
192 views

estimation of transition probabilities from aggregate data

Please, O mathematicians, help me understand the approach to the problem of estimating transition probabilities given only aggregate data in Kalbfleisch & Lawless' 1984 paper "Least-Squares ...
3
votes
3answers
115 views

Introduction to Markov Random Fields

I'm looking for a gentle introduction to this topic. The material I've found so far is substantially related to physics, and requires some background in such field. Is there anything more general and ...
3
votes
1answer
4k views

Kendall notation's “General distribution”, what does that mean?

The first and second parameters for the Kendall's notation may have a G value, which stands for General distribution, see here. But what does that mean? What is a ...
3
votes
1answer
57 views

Probability of a sequence of events in a Poisson process.

I am starting to study Poisson processes and I came up with this question: Let there be two Poisson processes with rates $\lambda$ and $\mu$ respectively, monitoring the occurrence of events (e.g. ...
3
votes
1answer
145 views

Applying equation to Markov process

This seems as an easy question, but however I can't handle it. In the following I need this fact: If $X=(X_t)$ is a Markov process with transition semigroup $(K_t)$ and initial distribution $\mu$ ...
3
votes
1answer
138 views

Levy processes with no positive jumps

Let X be a Levy process with no positive jumps and $\tau_y:=\inf\{t> 0: X_t > y\}$ then we have $$X_{\tau_y}=y\text{ on }\{\tau_y <\infty\}.$$ Could you explain that why? and does it hold ...
3
votes
1answer
139 views

What is the value of this game?

We have 3 black and 2 red balls in an urn. If we pick a black ball, we lose 1 USD. If we pick a red ball we win 1 USD. We can chose to start the game or not. If we start the game we can stop after ...
3
votes
1answer
97 views

Strong Markov property of Brownian motion

I was able to understand Brownian Motion $\{B(t):t\geq0\}$ has Strong Markov Property i.e. For any stopping time $\tau$, $P(B(t+\tau)\leq y | \mathcal{F}_{\tau})=P(B(t+\tau)\leq y|B(\tau))$ a.s. , $y ...
3
votes
1answer
339 views

In a continuous-time Markov process, is the waiting time between jumps a function of the current state?

Two books construct Markov processes from Q-matrices using waiting times and jump chains but differ in whether the waiting times depend on the current state. Can the two be reconciled? Klenke claims ...
3
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2answers
38 views

Always null recurrence at the boundary between positive recurrence and transience?

I have the following theorem: Let $\rho$ be the traffic intensity. a) If $\rho<1$, then $X$ is positive recurrent. b) If $\rho>1$, then $X$ is transient. c) If $\rho=1$, ...
3
votes
1answer
275 views

Elementary proof of geometric / negative binomial distribution in birth-death processes

The birth-death process concerns a population of $n_0$ individuals, each of which reproduce and die at a constant rate as time $t$ increases from $t=0$. Each individual splits into two individuals ...
3
votes
1answer
87 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 ...
3
votes
1answer
114 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 ...
3
votes
1answer
76 views

Random walk where increment depend on current position

Consider the following stochastic process, $$b(i+1) = b(i) + \xi_i (b_i),$$ where $\xi_i(b_i) \in \{-1, k \}$ are the independent increments having the following distribution: $$\begin{align} P (\xi ...
3
votes
1answer
65 views

For a generator $G$ of a Markov process in continuous time and finite state space, how would one prove that the entries of $e^{tG}$ are non-negative?

I have a generator matrix G for a Markov chain in continuous time and finite state space and I am looking to prove that the entries of $e^{tG} \geq 0 $ By definition $G = P'(0)$ with entries $g_{ij} ...
3
votes
1answer
107 views

Limit of a probability regarding a random walk

Consider a random walk on the integers starting at 0, where in each step you move either 1 or 2 meters (back or forth alike). As soon as you reach either the $N$ or $N+1$ meter mark, you stop. What is ...
3
votes
1answer
98 views

Markov property w.r.t. a countable state space

Background Let $\left(X_t\right)_{t \in I}$ ($I\subseteq\mathbb R$) be an $E$-valued stochastic process ($E$ being a Polish space with the Borel $\sigma$-algebra $\mathcal{B}\left(E\right)$) equipped ...
3
votes
2answers
546 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 ...
3
votes
1answer
535 views

Multidimensional infinitesimal generator of a jump-diffusion

Let $X=\{X_t\}_{t\geq0}$ be an $n$-dimensional Markov process, defined by the SDE $$dX_t = \mu(t, X_t) \, dt + \sigma(t,X_t) \, dB_t+\beta(t-,X_{t-}) \, dN_t,$$ where $\mu, \sigma$ and $\beta$ are ...
3
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2answers
66 views

Interpretation for the determinant of a stochastic matrix?

Is there a probabilistic interpretation for the determinant of a stochastic matrix (i.e. an $n \times n$ matrix whose columns sum to unity)?
3
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1answer
77 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 ...
3
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0answers
137 views

Generalization of Dobrushin's Ergodic Decomposition for continuous Markov Chains

Let $T$ be the shift transformation. Let $P$ be invariant for $T$ and also define a discrete state space Markov Chain. Let $C_{1},\ldots,C_{n}$ be the connected components of the Markov Chain. It ...
2
votes
2answers
235 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 ...
2
votes
2answers
56 views

Question on MIT Markov Matrices video

Markov matrices are pretty new to me and I'm a little rusty with my linear algebra. My question stems from watching this video from YouTube on Markov matrices. For those who wish to skip the video, ...
2
votes
1answer
114 views

A question about a stochastic process being Markov

Let $(X_{s},\mathcal{F}_{s})$ be a stochastic process adapted to a given filtration. I was told that, in order to prove that $X$ is Markov, it suffice to prove that for any nonnegative, ...
2
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1answer
39 views

Markov processes on function spaces

Is there any reference on Continuous time Markov process whose state space is infinite dimensional function spaces, such as the space of continuous functions $C(R^d)$? It seems Dirichlet Form is a ...