# Tagged Questions

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### 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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### 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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### 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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### 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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### Calculation of the Gallavotti-Cohen fluctuation theorem made by Lebowitz

I have a problem understanding a calculation in this paper (another form of the theorem an be found here at equation 11). For those who want to read the paper, I have difficulties with formula 2.14 in ...
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### Library chain stationary distribution

This is an exercise 1.47 from Richard Durrett's Essentials of Stochastic Processes p.85 (doi: 10.1007/978-1-4614-3615-7_1 or Google Books). On each request the ith of the $n$ possible books is the ...
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### Poisson Process (Easy)

I'm stuck at the following question: Customers with items to repair arrive at a repair facility according to a Poisson process with rate λ. The repair time of an item has a uniform distribution on ...
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### Is this two dimensional Markov chain correct for this queueing system?

The problem that I have two single server station with no queuing space a customer goes to station 1 if it is available else it goes to station 2 if it is available or it will be lost output from ...
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### Strong Markov property given transition functions

Suppose we are given family of transition functions satisfying Chapman-Kolmogorov equation, what conditions will ensure that there exists a continuous or cadlag Markov process with given transition ...
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### Stationary distribution for a Markov chain which is not irreducible

I have a Markov chain with $K$ states $S$: {$s_1,s_2,...,s_K$}. $s_1$ is reachable from any state in $S$; however not all the states can be reached from $s_1$. What does the stationary distribution ...
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### Does time homogeneity imply strong Markov property in a Markovian process

Does a time homogeneous Markovian process necessarily have strong Markovian property? Does continuity in state space, time, or path make a difference? What are the examples if it does not?
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### An example of a simple ergodic diffusion process?

I'm looking for a simple example (ideally two-dimensional) of an ergodic diffusion process with polynomial drift vector and diffusion matrix for which there a no known explicit expressions for the ...
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### Examples of decreasing-in-some-time-interval variance of a time homogeneous Markovian process

Let $x_t$ be a zero mean, time homogeneous Markovian process over time $t$ starting from $x_0=0$. What are the examples of $x_t$ where the variance at $t$ decrease over some interval of $t$? The ...
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### Under what circumstance should I use a Continuous time Markov Chain instead of a discrete time Markov Chain?

Why should I use one over the other, if I can basically reduce the small time-interval $h$ to be small enough that it simulates continuity? I guess this question is somewhat analogous to control ...
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### Transition function is a Markov semigroup?

How does the transition function in a Markov process become a Markov semigroup in time homogeneous Markov processes? Thanks a lot.
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### 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 ...
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### Show that all Gauss-Markov (GM) Processes can be Written in Recurrent Form

The book shows that the following is always GM. $$x_{k+1} = A_k x_k + w_k$$ where $$x_0, w_o, w_1,...$$ are independent and Gaussian. It then leaves the converse as "an exercise." Given: $$x_k$$ is ...
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### How is a simple birth process is time-homogeneous?

Why is it that a simple birth process is time-homogeneous? The incidence of a birth in a small time interval depends on the population size at the time of start of the interval. Doesn't this ...
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### Joint density of Markov process as product of conditional densities.

Let $(X(t))_{t\geq0}$ be a Markov proces such that s $X(0)=x_0$. Consider the random vector $(X(t_1),\dots,X(t_n))$ with corresponding joint density $g(x_1,\dots ,x_n)$. Is it then true that ...
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### Why is this infinite-state-space Markov chain positive recurrent?

Given the following transition matrix for a Markov chain, how can I see that the chain is positive recurrent? I want to convince myself that the chain has a limiting distribution, and the chain is ...
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### How to understand Markov property?

I'm learning stochastic process in college. How to understand Markov property?I'm curious about what is the power and validity of Markov property ? A stochastic process has the Markov property if ...
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### 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, ...
I have a question about Markov Process: Let $X_t=(X_t^1, X_t^2,..., X_t^n)$ be a Markov process with regard to the filtration $\mathcal{F}_t$, let $Y_t:=\max_{1\leq k\leq n}X_t^k$, then is $Y_t$ a ...