Tagged Questions

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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1answer
37 views

Why is the Stochastic Process in the HJM model non-Markovian?

I want to understand exactly what my title asks "Why is the Stochastic Process for the short rate in the HJM model of interest rates non-Markovian?" That process is the following: ...
1
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1answer
33 views

Connections between SDE and PDE

I have encountered a number of situations where the solution of a PDE and a certain expectation associated to a Markov process are equal. Two examples include: The heat equation $u_t = \Delta u$ ...
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0answers
32 views

general state space markov chain limit problem [on hold]

Suppose that the general state space $\chi$ is partitioned as $S$ and $S^c$ and $P(x, S^c)>0$ for any $x\in S$. How can one show that $P_x(\tau_{S^c}<\infty)=1$? I know how to show it when ...
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1answer
98 views

Brownian Motion inequality (related to Dvoretzky-Erdoes test)

i have the following question: Let $B(t)$ be a d-dimeansional Brownian motion $d\ge 3$, and $f$ be a monoton increasing function from the positive reals to the positive reals. Let $A_n=(\exists t\in ...
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1answer
17 views

Brownian Motion and Progressive Process

Let $B_t$ be a Brownian motion. Define sign function as follows. $sign(0) = 0$ and $sign(x) = \frac{x}{|x|}, \forall x \neq 0$. I do not know how to show the following two questions, especially on the ...
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0answers
24 views

Stationary distribution of a birth-death model where a parameter follows a uniform distribution.

I asked this question about some type a markov process I was interested in. @Did offers an answer but I fail to understand how to apply his answer to a concrete example. I am therefore seeking for an ...
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1answer
34 views

Application of Lévy–Khinchine formula

How can we express the characteristic functions of Wiener and Poisson processes by using the Lévy–Khinchine formula? I don't know how to find the characteristic functions of particular Levy ...
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4answers
259 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 ...
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0answers
49 views

Can we find a correlation between states of a Markov chain?

I have a fair bit of knowledge on Markov chains but I recently wondered if there is a way to find out a correlation between the states of a finite Markov chain. I could not find any material on this. ...
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0answers
10 views

Application of Strong Markov Property

Theorem SMP (Strong Markov Property) Let $X$ be a time homogenous Markov process with $T=\mathbb R_+$ or $\mathbb Z_+$ and let $\tau$ be a stopping time taking countably many values. Then ...
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0answers
43 views

Linear Filtering Problem (Keynman Fac/Particle Model)

$lienar Filtering Problem $$X_n^1 = X_{n-1}^1 + \epsilon_n *W_n $$ $$X_n^2 = (1-\alpha* \delta) X_{n-1}^2 + \beta*\delta X_n^1 $$ $$X_n^3 = X_{n-1}^3 + \delta*X_n^2$$ above is $$\approx$$ $$dX_n^1 ...
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1answer
21 views

Ergodic behaviour for bounded random dynamical system

Considering an iterated system described by $$ X_n =\gamma_nX_{n-1} , $$ where $\gamma_i$ are non-negative i.i.d. variables. It is easy to show that the expectation will grow unbounded for $\mathbb ...
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0answers
18 views

Stationary VS. limiting probability

I'm just wondering what the difference between stationary probability and limiting probability is. And, if any of you know: What does it mean that some elements exist and some elements doesn't, when ...
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0answers
28 views

Stochastic process using Markov chain (thief on the run!!)

I'm given an exercise where we are to simulate a thief escaping from an officer. The thief (let's call him/her T for simplicity) and an officer (O) have four cities to be in. Let's call the cities A, ...
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0answers
28 views

Covariance of states of a finite Markov chain

I know it is possible to construct a covariance matrix for states of a Markov chain but I cannot seem to find a proper way to compute it. I will attach some theories I found from Kemeny and Snell's ...
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0answers
128 views

Hidden Markov Model, transition probabilities which are modeled with an exponential distribution

I'm looking at implementing an algorithm described in a paper, but I'm having trouble understanding how the transition probabilities for a Hidden Markov Model are defined. In the first sections, I ...
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0answers
24 views

Strong Markov property of continous time Markov process

In the book "Applied probability and queues" which is available here http://books.google.de/books?id=BeYaTxesKy0C&pg=PA32&hl=de&source=gbs_toc_r&cad=3#v=onepage&q&f=false , ...
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1answer
57 views

Including non-markovian processes in a birth-death process

Current model I want to model a stochastic system with a birth-death (Markovian) model. I therefore have this kind of $n$ times $n$ (where $n$ is the number of possible states) transition matrix: ...
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0answers
8 views

A simple question on the inverse z-transformation of $\frac{z}{1-z}\mathscr{T}(z)\mathbf{q}$

I'm wondering if anyone who is familiar with the book Dynamic Programming and Markov Processes by Ronald Howard or simply z-transform can help me figure out an inverse z-transformation on page 23 of ...
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1answer
16 views

Recurrence of states in a function of a Markov chain

Suppose $X$ is a Markov chain (or process, for that matter) and suppose further $f(X)$ is also a Markov chain. Let $s$ be a recurrent state in $X$. Is there a general way to determine the recurrence ...
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1answer
33 views

Probability transition matrix for maximum of iid random variables

I have a homework problem that goes as follows: Let $\xi_i, \ i=0,1,2,\ldots$ be i.i.d. random variables of discrete type. The distribution of $\xi_0$ is given by: $$\mathbb{P}\{\xi_0=i\} = a_i, \ ...
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1answer
20 views

A certain formulation of the Chapman-Kolmogorov equation.

I am reading a book by Taira called Semigroups, Boundary Value Problems and Markov Processes. It is a nice read, but there is one thing I don't understand regarding the Chapman-Kolmogorov equation. A ...
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0answers
8 views

Finding a discrete Kalman-type process that produces a given Frequency spectrum

Given a power spectral density from f = -1/2 .. 1/2, is it possible to find a 1st order process that produces this series? In other words, x_i+1 = G x_i + W r_i ...
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1answer
28 views

Show that this Markov chain is recurrent

So I have a Markov chain on the nonnegative integers such that, starting from $x$, the chain goes to $x+1$ with probability $p$, $0<p<1$, and goes to state $0$ with probability $1-p$. I'm ...
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1answer
11 views

Markov Processes: $P_x$ and $E_x$

In the study of Markov processes, one usually introduces the measures $P_{\pi}$ on the path space of the process where $\pi$ is an initial distribution of the process $X$ i.e $\pi=\mathcal L(X_0)$. ...
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0answers
10 views

Equivalent Formulation of Markov Property for Homogeneous Chains

In Shiryaev's Probability (just above the strong Markov property, p.568), the author says that an equivalent formulation of the usual Markov property for homogeneous chains is $$P[\theta_nX\in B\mid ...
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0answers
17 views

Strong Markov Property for Discrete Stopping Times

I'm having a hard time deciphering a particular proof of the following strong Markov property. Theorem (Strong Markov Property) Let $X$ be a time homogenous Markov process with $T=\mathbb ...
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1answer
19 views

The “on $\left\{ \tau <\infty \right\}$” in the Strong Markov Property

The strong Markov property is often formulated as $$P[\theta _{\tau}X\in A\mid \mathscr F_{\tau}]\overset{\text {a.s on }\left\{ \tau <\infty \right\} }{=}P_{X_\tau}(X\in A)$$ What exactly does ...
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2answers
37 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)?
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1answer
33 views

Is this a Markov chain? [duplicate]

Let $\{\xi_n \}_{n \geq 1}$ be i.i.d random variables taking values on $\mathbb{Z}$. Let $\xi_0 = 0$. $S_n = \sum\limits_{i=1}^{n} \xi_i,$ where $S_0=0$ $Y_n = \sum\limits_{i=0}^{n} S_i$. My ...
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0answers
16 views

Markov vs reinforcement learning

What's the different between markov chain ,markov decision process and reinforcement learning? when we can apply these theories?
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0answers
16 views

markov process with extra boundary

In a markov process a random walker has to reach N (absorbing boundary) from $x_o$ on a $[0,N]$ lattice, where $0$ is the reflecting boundary. To find the first exit time of the random walker via N, i ...
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0answers
16 views

Simple random walk conditioning on non-return

Consider a simple symmetric random walk on $\mathbb{Z}$, $(S_t)_{t \geq 0}$, with $S_0=0$. Let $P_{k,j}$ be the probability that the walker hits the point $k$ without returning to the origin in ...
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1answer
36 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 ...
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1answer
56 views

Markov Chains : Can anything be said about what happens in between two transition?

In time homogeneous discrete Markov chains we take a set period for a single transition. In examples we see sometimes depending on the examples the transition period being a a month a week etc. I'm ...
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1answer
99 views

A Markov Chain Problem.(Change the color of ball)

There are $n$ different color balls in a box. Take two balls in turns, and change color of the second ball to the first. (This is one operation). Let $k$ be the (random) number of operations needed to ...
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0answers
33 views

Parental Markov Condition Example

I'm currently reading a text on Bayesian networks and the text is giving some very crude interpretations of what appear to be some of the most important foundations of the subject. It states the ...
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3answers
103 views

Transition Matrix of M/M/1 Queue

We know that for an M/M/1 queue the state space is $S=\{0,1,2,... \}$. Further the probability to go from state $i$ to $i+1$ is $\lambda$ for all $i$ in $S$. Moreover, to go from $i$ to $i-1$ is the ...
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0answers
56 views

Follow-up on solution to Markov process equation

I asked a question here about solving a system related to an absorbing Markov chain. I now have a variation where there are $m$ types (of student, job seeker, etc) each of which applies to ...
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0answers
16 views

Markov Models and Applications

I am looking for resources in Markov models and its applications. I'm looking for tutorials, videos, books etc which provide the following Explain Markov chains in layperson terms and provide ...
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1answer
49 views

Is there a solution to this system for the diagonal matrix?

I'm trying to find a solution to a system of equations, but its quite different from anything I've come across before. I believe there is a solution, but I could be wrong. $\mathbf{A} = ...
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1answer
30 views

Markov property for a stochastic process with discrete state space.

Consider a stochastic process $\{X_s\}_{s\in\mathcal S\subseteq\mathbb R}$ with value in $(\mathbb R,\mathcal B(\mathbb R))$ adapted to a filtration $\{\mathcal F_s\}$ (we can suppose that ...
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0answers
15 views

What are the techniques to solve finite-horizon Markov decision process with large yet finite state and action space?

I formulate a problem into a Markov decision process with finite horizon and plan to solve it with the Backward Induction algorithm. However, both the state and action space are large (on the order of ...
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0answers
23 views

How to solve “So Who's Counting” problem using Markov Decision Process?

In Martin Puterman's book Markov Decision Processes, one of the problems he gives is "So Who's Counting". In that problem, 5 random digits are generated. After each digit is generated, it is placed in ...
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0answers
18 views

Probabilities in a Markov Model

I am reading a paper on Markov Models and I am trying to figure out how to compute the probabilities for the $\alpha$-pass. I am given an $N\times N$ matrix $A$, that has the probabilities of ...
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0answers
29 views

Treatment of Markov process with absolute states

In the standard treatment of a markov process, the state vector is a probability vector, whose elements can be between zero and one. But I have a need to constrain the state vector to zeros or ones. ...
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0answers
40 views

Exact probability distribution for hitting time of simple random walk

Consider simple random walk on the line starting from the site $y \in \mathbb{N}$. With probability $p$ the walker moves to the right and with probability $1-p$ to the left. Call $\tau$ the first time ...
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1answer
48 views

Mean absorption time for pure birth process

Let $\xi_t$, $t\geq0$, be a pure birth process, with $P\{\xi_{t+h} = i +1 | \xi_t = i\} = \lambda^ih + o(\lambda)$, as $h \downarrow 0$. At $t=0$, $\xi_0 =1$. Let $\tau = \min\{t ~|~ \xi_t = N\}$. ...
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1answer
34 views

Proving that a process has the Markov property

Let $X_t=xe^{ct+aB_t}$ where $B_t$ is one dimensional Brownian motion. How would I prove this is a Markov process using the expectation definition of a Markov process, i.e., ...
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1answer
29 views

A question about Markov process

Let a stochastic process $(x(t),\theta(t))$ be given by $$ \dot{x}(t)=f(x(t),\theta(t)) $$ for a well defined continuous function $f(\cdot,\cdot)$. Let $\mathcal{F}_t$ denote the natural filtration ...