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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A Markov Chain probability, conditioned on a random time.

My question: Upon reading theory about diffusion processes, i came across an argument which i believe simplifies to this: Say we have a Borel measurable set $A$ (if it matters you can set $A=\lbrace ...
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Confused about definition of absorption probability

My confusion can probably most easily be explained with an example. Consider the following one step transition matrix : $$ P=\matrix{% & 0 & 1 & 2 & 3 & 4 \\ 0 & ...
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Reference for General state space Markov chain

What is a good reference for general state space Markov chains? Is there a reference which assumes only familiarity with finite/countable state space Markov chains and then extends the results (e.g., ...
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38 views

Why is $f(X_t)-\int_0^t Af(X_s) \, ds$ a martingale for a Markov process $(X_t)_{t \geq 0}$?

I think if $A$ is the usual generator for the Markov process $(X_t)_t$ $$A f (x) = \lim_{t \downarrow 0} \frac{\mathbf{E}^{x} [f(X_{t})] - f(x)}{t}$$ then we get that for any "nice" $f$ the process ...
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1answer
47 views

Understanding the Markov property of Brownian motion

I'm trying to understand the Markov property for Brownian motions in full generality. The textbook I'm following states it like this: Recall that we have a family of measures $P_x, x \in ...
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Expected response time of Continuous time Markov chain

I'm studying CTMC (Continuous Time Markov Chains). I came across the following slide I don't understand how they got $M(t+h) = M(t) + \alpha h + M(t)\lambda h - M(t) \mu h +o(h)$ Could anyone ...
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Markovian systems: Why must controls be independent of state?

I am currently working my way through Probabilistic Robotics by Thrun, Burgard, and Fox. On p. 91, I encountered the following statement: The Markovian assumption implies independence between ...
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12 views

Joint Markov Chain (Two Correlated Markov Processes)

I have two Markov Chains, and they exhibit some correlation between them. For instance, when Chain A moves to state i, there is a high likelihood that Chain B moves to state j. How would I go about ...
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40 views

Defining a function over time in terms of a random variable that is undefined at a certain time

Let $X_n$ be a random variable taking on one of three values $a,b$ or $c$ over time. That is, for each $n \in \mathbb{N}$, we have $X_n \in \{a,b,c\}$. Also, for each $n \in \mathbb{N}$, let $F_n$ be ...
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36 views

Why is the stationary distribution a distribution?

Suppose we have a time-homogeneous, discrete-time, aperiodic, positive recurrent, irreducible Markov chain $(X_t)_{t \geq 0}$ on a discrete state space $E$. It is known that its stationary ...
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84 views

How to prove a set is a core for an infinitesimal generator

I am trying to prove that the set $D=\bigcap_{n\geq 1} \{f\in C^2 (\mathbb{R}) :f(0)=\sum_{k\geq 0} f(\frac{k}{\sqrt{n}})g_n (k)\}$ is a core for the infinitesimal generator of reflecting brownian ...
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1answer
27 views

Build a Markov process from a transition semigroup

Suppose that we have a Markov process $X$ with time-homogeneus transition probability function given by $p(t,x,A)$, with $t>0, x\in E, A \in \mathcal E$, where $(E,\mathcal E)$ is the state space. ...
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Moment Generating Function for Brownian motion's exit of interval.

Let $B(t)$ be a standard BM. Consider the stopping time $T = \inf\{ t > 0: |B(t)| = a\},$ the usual first exit time of the interval $(-a, a).$ We can see that $\mathbb{E} e^{tT} < \infty$ for ...
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1answer
20 views

A question involving Markov processes

Let $(S, \mathcal{B}, m)$ be a measurable space and $X_p := L^p(S, \mathcal{B}, m)$. Let $T_t \in \mathcal{L}(X_p, X_p)$ be a bounded linear operator defined by $$(T_t f)(x) = \int\limits_S P(t, x, ...
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35 views

Krylov-Bogoliubov theorem in discrete and continuous time

Consider the following two similar frameworks (the first one in discrete time, the second one in continuous time) in which I am trying to apply the Krylov-Bogoliubov theorem. 1) (Discrete time.) ...
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1answer
34 views

Distribution Stopping time under Brownian motions

Considering $W$ the canonical process on $C([0,1],\mathbb{R})$ and the row filtration generated by the coordinate process of $W$, I want to prove that ...
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45 views

Semigroup associated to a Markov process

I'm studying the transition semigroup associated to a Markov Process, in particular the Hille-Yosida theorem and the Martingale Problem. In my notes I found : "If $\{T_t\}_t$ is a strongly continuous ...
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1answer
39 views

Properties of a transient state in a Markov Chain

I have been trying to solve this problem for a while now Prove that if $j$ is transient state, then $\displaystyle\sum_{n=1}^\infty p_{ij}^{(n)}<\infty \ \forall i \in S$, with $S$ the state ...
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1answer
31 views

Proving technique used to show an equivalence to the definition of a Markov process

Let $X=(X_t)_{t\in I}$ be Markov process with values in a Polish space $E$. I want to show, that there exists a stochastic kernel $\kappa:E\times\mathcal{B}(E)^{\otimes I}\to [0,1]$ such that ...
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1answer
47 views

Assume a die is rolled repeatedly. Find the markov matrix $P$ for the random variable of the time until the next $6$.

Assume a die is rolled repeatedly. Find the markov/transition matrix $P$ for the random variable $X_r$ = the time until the next six at time $r$. My solution was: For $i,j \geq 0$, $P$ is given ...
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69 views

Markov Chains - Strong Markov Property

I'm struck with an exercise. I tried, but the results don't seem to fit to those proposed. Exercise: Two players play the following game. The one who begins draws two cards from a deck of 40 cards ...
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1answer
72 views

Autocorrelated, discrete, bounded and symmetric random walk with no edge attraction

I need to move over a discrete set of linearly organized.. let's say "Japan steps" $S=\{0,\dots,c\}, c \in \mathbb{N}^*$. My current position is given by $d \in S$. On each time step, I need to draw ...
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18 views

Factorization of the Fokker-Planck semigroup

"In the classical theory of Markov processes, the Fokker-Planck semigroup $\{T_t:t\ge 0\}$ can be factored as $T_t=N\circ U_t \circ j$, $t\ge 0$, where $j$ is an embedding, $U_t$ is a group of ...
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1answer
37 views

Markov processes and $C_0$-semigroups

A Markov process $(X_t)_{t \geq 0}$ in continuous time on $\mathbb{R}^d$ can be described by a semigroup of Markov kernels $(p_t(x,A))_{t \geq 0}$ with $p_0(x,A) = 1_{A}(x)$ and which fulfill the ...
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26 views

Simple Markov property

I want to prove the simple Markov property but I come to a point where I do not see how to conclude. I want to prove $\mathbb{E}_\nu[Z\circ\Theta_t\mid ...
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16 views

model a system with finite users as a Markov Chain

I have to model a system M/M/2 with finite users (4 users) as a Markov Chain (and then find the probality an incoming users would enter the queue being the servers busy but that's not the problem). I ...
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7 views

Looking for resources on Harris recurrence

I'm working on a problem (in a not countable space) and it seems that I could get much further with it if I can prove that a certain Markov chain is Harris recurrent (I strongly suspect that it is). I ...
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1answer
26 views

How to understand this kind of Markov chain?

There are two unidirectionally coupled processes $X_t$ and $Y_t$. The coupling is $Y_t=g(X_{t-\delta},\dots)$. The Bayesian network of the two process is described in the following figure: Now this ...
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10 views

rigorous definition of adjoint infinitesimal generator for Markov process?

A Markov process on a $\sigma$-addive measurable space $(E,\Sigma,\mu)$ can be described by a family of operators $(P_t)_{t \geq 0} ,\, P_t: L^\infty(\mu) \to L^\infty(\mu)$. These operators define a ...
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Markov Chains and transition semigoups

I'm trying to figure out what the following statement refers to. A process $X$ is markov with transitions semigroup $(K_t)_{t\geq0}$ and initial distribution $\mu$ if and only if for all ...
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14 views

How does one estimate the order of a Markov chain empirically (given the data)?

I have a string of symbols $x_1, x_2, ...., x_n$, ($n$ very large), belonging to a finite alphabet. I know that they are a result of a Markov process, but I want to find out the order of the process. ...
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Correct steps in rewriting expectation to a probability

My knowledge of measure theory and probability spaces is limited, so please keep it relatively simple. Let $\{X(t), ~ t \ge 0\}$ be a Markov process on the countable state space $\mathbb{N}_0$ with ...
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Electrostatic capacity of two spheres with changing radii

Although I have read a lot of questions and answers here, this is my first time actually posting. Feel free to suggest needed edits. My question is the following (in a simplified setting). All this ...
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Doubt Concerning Markov Property

Given a Markovian process $(X_t )_{t\geq 0 }$, is the following property accurate? $$\mathbb E \left[ f(X_{t_1}, X_{t_2},X_{t_3}) \mid \mathcal F ^X_{t_2}\right] = \mathbb E \left[ f(X_{t_1}, ...
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1answer
24 views

Condition for the existence of a Markov process from the properties of a semigroup $\{T_t\}$

In the article Diffusion processes with continuous coefficients I (1969, Stroock and Varadhan) one finds the following arguments in pages 26-27 "$(\cdots)$ for any $ \epsilon >0, \sup_{x \in ...
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41 views

Calculating the generator of a weighted transition function

Let $(P_t)_t$ be the transition function of a Feller-Dynkin process $X$. The usual Banach space of functions that the semigroup $(P_t)_t$ is working on is $C_0(E)$, i.e. continuous functions that ...
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17 views

Ergodicity property for continuous-time Harris positive Markov process

The following theorem is Theorem 13.3.3 of Meyn and Tweedie's Markov Chains and Stochastic Stability on page 328 Theorem 13.3.3. If $\Phi$ is positive Harris and aperiodic, then for every initial ...
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1answer
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Expression for the quotient between two stationary states in a Markov process

I've been thinking about this problem and I would appreciate some help. Consider a finite number of states ($n$) Markov process with transition matrix $Q_{n\times n}$ with the usual properties and ...
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27 views

Probability of hitting zero

Suppose time is discrete. $X_{t+1} = X_t + x_t$. $x_t$ is of continuous value, iid with mean zero and finite variance. Let initial condition $X_0>0$, how can I prove that the probability of $X_t$ ...
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31 views

Can solution to Rubik's cube be seen from the point of view of Markov Decision Process?

Solving Rubik's cube can be thought of as a Planning problem which has : a state space $S$ a set $G \subseteq S$ of goal states (in this case singleton) actions $A(s) \subseteq A$ applicable in ...
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1answer
55 views

Is this PMF or PDF?

I am reading a technical report on expectation-maximization (EM) algorithm (http://melodi.ee.washington.edu/people/bilmes/mypapers/em.pdf) and I am confused about something. For HMMs, it defines ...
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Discrete-Time Stochastic Calculus and Stopping Times: Resources

In my measure-theoretic probability course we covered what the professor called "discrete-time stochastic calculus". Essentially, it was a three part method for computing certain quantities such as ...
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95 views

Server farm model and non-explosiveness

Consider the following server farm model. Customers arrive at a server farm according to a Poisson process at rate $\lambda$, each requesting a machine for an amount of time that has an exponential ...
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1answer
55 views

The probability of going bankrupt

The question looks like a gambling problem. But I am not sure whether it is the same or similar to gambler's ruin problem. Assume I have wealth $W_t$. At each step $t$, I encounter a random shock ...
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21 views

Partial differential equation involving a random process (literature advice)

In articles like this one (end of page one and page two), physicists often tend to treat a random process with discrete time and countable space set as a differentiable function (whose domains are ...
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1answer
19 views

Ornstein-Uhlenbeck-Model with irregular time intervals

I have data of the form: Time t Price x(t) 0 80 21 82 24 82.3 32 81.5 ... ... The point is, that the time intervals are highly ...
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1answer
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Simple random walk: What is the probability that the hitting time is exactly 2n?

I refer to the random walk $(S_n)_{n \geq 1}$ where $S_n = X_1 + \cdots + X_n$ and $X_i$ are i.i.d random variables taking values $\pm 1$ with equal probability. I want to know how to show that ...
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25 views

Limit of decreasing sequences of markov time (stopping time) is markov time?

Let $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t \geqslant 0}, \mathbb{P})$ be a filtered probability space and let $\tau_n \geqslant \tau_{n+1}$ be a markov time (stopping time) with respect to ...
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Discrete time Pure-Birth process with population fraction

I would like to solve difference equations for a pure-birth process, where the rate of adding new nodes depends on the fraction of population. $$x_i(t+1)-x_i(t)=\alpha ...
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What is the axiomatic definition of potential kernel?

In the enter link description here, a kernel V is a Markov potential kernel if the operator I+V is invertible and its inverse is of the form I􀀀-N with N a sub-Markov kernel.The definition of Markov ...