Tagged Questions

0
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0answers
22 views

On discrete-time stochastic attractivity of linear systems

Let $m$ be a probability measure on $Y \subseteq \mathbb{R}^p$, so that $m(Y)=1$. Consider a continuous function $f: \mathbb{R}^n \rightarrow \mathbb{R}^p$. Assume that $f(0) = 0$, and that there ...
0
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1answer
46 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
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2answers
78 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 ...
-2
votes
2answers
83 views

Is first order moving average a Markov process?

Given first order moving average $$ x(n) = e(n) + ce(n-1) $$ where $e(n)$ is a sequence of Gaussian random variables with zero mean and unit variance which are independent of each other, and $c$ is ...
0
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3answers
247 views

Finding the transition probability matrix, two switches either on or off..

Each of two switches is either on or off during a day. On day n, each switch will independently be on with probability [1+number of on switches during day n-1]/4 For instance, if both switches are on ...
3
votes
1answer
128 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 ...
5
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0answers
77 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 ...
1
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1answer
66 views

Is this Markov?

Consider a process $\{X_n, n\geq 0\}$ with state space $S=\{0,1,2\}$ s.t. $$ P(X_{n+1}=j | X_n=i, X_{n-1}=i_{n-1}, \dots, X_0=i_0)=\begin{cases} P_{ij}^I \ \ \ n \ \mbox{ even},\\ P_{ij}^{II} \ \ \ ...
1
vote
1answer
78 views

Why is the following example of a Markov process not strong Markov

$X(t) := 0 \;\; (t \leq \tau),\;\; t - \tau\;\;(t \geq \tau)$ with $\tau$ exponentially distributed. Then X has the Markov property but not Strong Markov Property. But why ???? Can someone kindly ...
1
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0answers
57 views

“To every Q-matrix corresponds a unique Markov process.” Proving uniqueness

"To every Q-matrix corresponds a unique Markov process." I'm trying to understand Klenke's proof of the uniqueness part of this proposition. Klenke's proof Following is an adapted version of ...
2
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1answer
126 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 ...
1
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0answers
32 views

To every Markov process corresponds a Q-matrix? [duplicate]

Possible Duplicate: Logarithm of a Markov Matrix It is known that to every Q-matrix corresponds a unique Markov process. Does the converse hold? Specifically, Can a (discrete state, ...
1
vote
1answer
109 views

To every Q-matrix corresponds a unique Markov process

"To every Q-matrix $q$ corresponds a unique Markov process." I'm trying to understand Klenke's proof of the "existence" part of this proposition, namely that given a Q-Matrix $q$, there exists a ...
1
vote
2answers
91 views

A Poisson process's distributions

The Poisson process $\left(N_t\right)_{t\in\left[0,\infty\right)}$ is supposed to be a Markov process, but a Markov process $\left(X_t\right)_{t\in I}$ should be coupled with a family of distributions ...
2
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1answer
143 views

Showing that the Poisson process is characterized by five properties

Klenke defines the Poisson process as a family of non-negative integer valued random variables with independent increments, whose increments are distributed according to the Poisson distribution ...
0
votes
1answer
69 views

The strong Markov property with an uncountable index set

The following definition of the strong Markov property, from Klenke's book, supposes an index set $I$ that is not necessarily countable. However, it is explicitly mentioned previously (following Lemma ...
2
votes
1answer
109 views

Prove that Brownian Motion absorbed at the origin is Markov

I'm trying to prove that Brownian motion absorbed at the origin is a Markov process with respect to the original filtration $\{\mathcal{F}_{t}\}$. To be more specific, let $(B_{t},\mathcal{F}_{t})_{t ...
2
votes
1answer
228 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. ...
2
votes
1answer
72 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
votes
1answer
69 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 ...
2
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0answers
65 views

Equivalence of Markov Property

Suppose we are given an $E$ valued stochastic process $(X_t)_{t\in T}$. The time set $T$ is in this context equal $[0,\infty)$. Then we define the canonical realization on the path space $E^T$ as the ...
2
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1answer
625 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 ...
0
votes
1answer
154 views

How are the pairs of two independent pure-birth processes a Markov process?

A pure-birth Process is a generalization of a homogeneous Poission process. Whereas in the Poisson process the holding times between jumps are iid exponentially distributed random variables with ...
4
votes
1answer
122 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 $\mu$ ...
2
votes
2answers
116 views

Markov chain basic positive recurrency question

If a discrete markov chain is stationary (as far as I know: doesn't modify itself with time), irreducible (doesn't have transient states) and aperiodic (no periodic states), is it positive recurrent? ...
2
votes
0answers
122 views

Is every killed Markov process still a Markov process?

Suppose we've got $X=(X(t))_{t\geq 0}$. $X$ is a strong Markov process with respect to filtration $\mathcal{F}_t$, taking values in some subset of $\mathbb{R}$. We take $\tau$ - a stopping time w.r.t ...
2
votes
1answer
1k 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 ...
1
vote
0answers
93 views

Continuous-time Markov process - need help with flux and discrete-time equivalent

I have a continuous-time markov process and I need to calculate the following transition frequencies matrix (aka intensity matrix) transition probabilities all parameters which define permanence ...
11
votes
2answers
354 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 ...
1
vote
1answer
184 views

How does this follow? Markov Chain and conditional expectation question.

I have the following from a book: Assume that $$ P_x(\tau_C \circ \theta_{(k-1)N} > N|F_{(k-1)N}) = P_{X_{(k-1)N}}(\tau_C > N). $$ Integrating over $\{ \tau_C > (k-1)N\}$ using the ...
2
votes
1answer
67 views

The property preserved under the perturbations of the kernel

For a measurable space $(E,\mathcal E)$ and a Markov kernel $P:E\times \mathcal E\to[0,1]$ there is a unique homogeneous Markov chain $X$. The first return time is defined as $$ \tau_A = \inf\{k\geq ...
3
votes
1answer
109 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 ...