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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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, ...
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125 views

Probability of a trajectory in Markov processes

I need help with a simple formula! (My question is taken from here, pag 26 eq 1.112. ) Consider a Markov Process with associated Master Equation: \begin{equation*} ...
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28 views

Poisson process different type of events

Suppose that it arrives people to a store according to a poisson process with rate $\lambda = 6$/hour , females arrive with probability $0.6$ and male with $0.4$. What is the probability that there ...
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Continuous time Markov chain. proportion of time spent in state i

If a question asks for the proportion of time spent in a specific state is this the same as the stationary distribution or something else? For continuous time Markov chain with finite state space.
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44 views

Continuous Markov chains, arriving pairs

I have been trying to sort out this exercise but really stuck on this. Preparing myself for exams and found many exercise on continuous Markov chains but I am always stuck when it comes to transition ...
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19 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 ...
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31 views

Markov chain property

I would like to make clarification and show my curiosity about markov process. I will show some part of definition related to markov process from here. The Markov property is the dependence ...
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A doubt on markov decision process

Given that a policy is a function from a state action pair to probabilities, the set of policies for a MDP forms a POSET (the partial order is due to value function for a policy). Why there should be ...
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1answer
29 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 $$\forall t\ge 0: M_t:=\max_{0\le s\le t} B_s$$ ...
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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 ...
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15 views

Policy Adjustment in Markov Decision Process

I was using MDP on my work to make optimal decision. I used discrete time, finite state MDP. I assumed that I will have an initial parameters, like the Reward/Cost, state transition probabilities and ...
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49 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 ...
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56 views

Inverse of a regular stochastic matrix

Is it true that the inverse of a regular stochastic matrix is also regular? Are there any other interesting features that the inverse may have of a regular stochastic matrix? Hope someone could answer ...
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59 views

General birth and death process

hi i need some help to understand the following (from the general birth and death process).I'll give some context first , then i ask questions. Consider general birth and death process with birth ...
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32 views

References for time-inhomogeneous Markov jump processes?

In some central models in life insurance mathematics, the state of the insured is modeled using a continuous-time time-inhomogeneous Markov process with finitely many states. While many results for ...
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36 views

relations between properties of stochastic processess

If we have an integer valued stochastic process, are these implications correct? independent increments $\rightarrow$ Markov property Markov property $\nrightarrow$ independent increments stationary ...
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1answer
56 views

time-homogeneous continuous time Markov chain

I have a question about the continuous time Markov chain. In the Poisson process we have independent and stationary increments. Do we have this in a continuous time Markov chain that is ...
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46 views

Markov Chain with Normal Transition Matrix

Consider a (sub)-stochastic matrix $P$, and the associated Markov chain $X$ with \begin{align*} \mathbf P [X_n =y | X_0 = x] = P_{xy}^n. \end{align*} Suppose we have the condition $P^T P = P P^T$, ...
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1answer
33 views

Why does markov chains power method converge at the rate of |λ_2/λ_1 |

I'm doing some researches on Markov Chains, and every time I meet this statement, that The rate of convergence of the power method is given by |λ_2/λ_1 |^k→0, when k→inf. And where λ_1 and λ_2 are ...
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19 views

Markovian Coefficients Unclear Definitiondtdt

I have come across the following unclear definition: Consider $dS(t) = S(t)[\mu(t)dt + \sigma(t) dW(t)]$ "Assume that the coefficient $\sigma$ is Markovian. That is, (with abuse of notation) ...
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24 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 $\lambda_c$ ...
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1answer
30 views

Markov process which is not martingale

I have seen the examples of a discrete time martingale that is not a Markov Process. Can you construct me an example of discrete time Markov Process that is not a martingale?
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20 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 ...
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11 views

Hidden Markov Model to Markov Model

Can every HMM be converted into some $nth$ order Markov model? My thoughts are that you could just multiply the emission probabilities in each state by the probability of state transitions. For ...
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23 views

Computing the PMF for N(t) in a renewal process

I'm in a stochastic processes course, and we just started on renewal theory. Unfortunately, we skipped the section on queuing theory, and nearly example in my textbook for renewal processes uses ...
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1answer
30 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 ...
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1answer
31 views

Continuous-time Markov Question

I have a question about a continuous-time Markov process on the discrete space. I am given the generator and asked for find the expected time the Markov process needs to get back to state 3, given ...
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1answer
35 views

Showing the square of a Markov process is or isn't Markov

Hi I am trying to show that if $X_n$ is a markov process, whether or not $X_n^2$ is a markov process. $X_n$ is a markov process if $P\{X_k = a_k|X_{k-1} = a_{k-1}, X_{k-2} = a_{k-2}, ..., X_k = a_1 ...
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2answers
57 views

Find the generator of Markov Process

Homework question: Consider the Markov process $X_t=B_t-t^2+t$ where $B_t$ is the Brownian motion. Find the generator $Q$ of this process. I am completely confused how to find the generator for ...
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1answer
27 views

Steady state of a $4 \times 4$ transition matrix

Normally I just take $q(M_{m\times n} - I_{m\times n})$ to workout the steady state, but here I have: $$\left(\begin{array}{rrrr} 0 & 0 & .8 & .2 \\ .4 & .6 & 0 & 0 \\ .2 ...
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1answer
23 views

Limiting probability of Markov chain(Terminology)

If I am asked to find the limiting probability of a Markov chain, what does this pertain to? $\lim \limits_{n \to \infty} P^n$? Where $P$ is the stepping matrix and $n$ is the number of steps. "What ...
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28 views

Error in thinking: Poisson Process is a Markov Process

I am a bit confused on proving the Markov property for Poisson processes. That is, we want to prove, if $X = (X_t: t \in \mathbb{R})$ is a Poisson process with rate $\lambda$: $P(X_{t_n} = a_n | ...
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49 views

How's the damping factor in Google PageRank algorithm calculated

I'm doing some researches about Google's PageRank algorithm for my thesis, I've found that the damping factor x (for example), where x is in : P` = x.P + (1-x)Q where P is the original ...
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49 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 $j = i+1$} \\ ...
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25 views

showing a condition implies convergence to invariant distribution

For an arbitrary Markov chain, I'm trying to show that $\lVert{ P_{\mu}\circ \theta_n^{-1} - \nu \rVert}\to 0$ iff $\sum_j \lvert{ P_{ij}^n - \nu(j)\rvert}\to 0$, where $\theta_n^{-1}$ is a shift ...
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22 views

inequality for finite state markov chains

Let $X$ be a discrete-time Markov process in $S$ with invariant distribution $\nu$. Show that for any measurable set $B\subset S$ such that $$P_{\nu}\{X_n \in B\, \textrm{i.o.} \}\geq \nu B.$$ I'm ...
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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 ...
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15 views

radius of markov chain

For an irreducible markov chain one can show, that $\limsup \sqrt[n]{p_{ij}^{(n)}}$ is independent of the choice of the states $i$ and $j$ where $p_{ij}^{(n)}$ is the probability to get from $x$ to ...
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44 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 ...
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1answer
75 views

Ornstein-Uhlenbeck processs: Markov, but not martingale?

I'm puzzled about properties of the Ornstein-Uhlenbeck process, given by the Itō integral $$ X_t = x e^{-\lambda t} + \sigma \int_0^t e^{-\lambda(t-s)} d W_s \,. $$ I compute that $\{X_t\}$ is not ...
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50 views

Markov chain with infinite number of transient and positive recurrent states?

Is it possible to have a markov chain with an infinite number of transient states, and an infinite number of positive recurrent states? Thank you!
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231 views

Prove markov chain is null recurrent

Two fair coins are tossed repeatedly. Let Xn denote (Total Number of Heads from Coin 1)-(Total Number of Heads from Coin 2) after n tosses. Thus the state space is {0, ±1, ±2, .... }. Show that the ...
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19 views

Three-State Markov Process with Differential Equations

This question is from a take home quiz and I could really use the help. Thanks in advance
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2answers
160 views

Mean recurrence time and stationary distribution of a Markov chain?

In a Markov chain is there a theorem relating the existence of the stationary distribution and the mean recurrence time? E.g. impossible for stationary distribution to exist therefore mean recurrence ...
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70 views

proving null recurrence of random walk (Markov chain)

How would I prove that the zero state of a random walk with a positive probability of staying in the same state is null recurrent. (sorry if this isn't a random walk and just a Markov chain.) eg. ...
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83 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 ...
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1answer
28 views

Show $P(S_{2n}=x|S_0=x) \ge \frac{1}{N}$

Let $X_n$ be an aperiodic, discrete-time Markov chain so $S=\{1,...,N\}$ whose transition probability is symmetric. How can I show that for all $x \in S$ and all integers $n$, $P(S_{2n}=x|S_0=x) \ge ...
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25 views

Maximal principle for elliptic or linear integro-differential operator

Consider $L$ the operator forming as $$ Lg= -g^{'}(x)+(g(x+1)-g(x)) $$. $h$ on $[0,\infty)$ satisfies the following integro-differential equation $$ Lh \geq 0 $$ with boundary condition: $$ ...
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1answer
35 views

Transition rate matrix from transition probability matrix

If I have a $2 \times 2$ continuous time Markov chain transition probability matrix (generated from a financial time series data), is it possible to get the transition rate matrix from this and if ...
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58 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 ...