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I have a continuous Markov Chain with transition matrix $\Bbb P$ and with initial state $X_0=1$ and state space $I=\{1,2,3,4,5\}$ $$\Bbb P= \begin{bmatrix} -3 & 1 & 0 & 1 & 1\\ 0 & -1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & -3 & 1 \\ 0 & 1 & 0 & 0 & -1 \end{bmatrix}$$

I have to find the discrete skeleton of matrix P,that in my exercise results:$$\begin{bmatrix} 0 & \frac13 & 0 & \frac13 & \frac13\\ \frac13 & 0 & \frac13 & 0 & \frac13 \\ 0 & \frac12 & 0 & 0 & \frac12 \\ \frac12 & 0 & 0 & 0 & \frac12 \\ \frac14 & \frac14 & \frac14 & \frac14 & 0 \end{bmatrix}$$

I searched a lot on internet a good definition and a way to compute the matrix ,but i find nothing.

Federico
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  • @JimmyR. No,i find it on an exercise gives us in a lesson,but i searched on internet a definition but i find nothing :( – Federico Jan 25 '16 at 15:27
  • Now the second matrix is a transition matrix... which corresponds to neither of the two discretizatons of the process whose generator is the first matrix that one could think about. – Did Jan 25 '16 at 15:35
  • I'm sorry if i am confused ,but this topic is fairly new to me , i just copied down the solutions that my TA gave in class,i don't know how to solve it and i would appreciate some hints on how to proceed.I didn't understand completely your last comment – Federico Jan 25 '16 at 15:43
  • The obvious move is to go back to your TA and ask them for the definition of skeleton in this context (that is, if said definition was not already given to you, which somehow I doubt). – Did Jan 25 '16 at 15:45
  • @Did Thank you.I would have done it , if the TA was avaible ,which is not.Could you provide some links where they explain this concept because i can't find it anywhere online. – Federico Jan 25 '16 at 15:51
  • See https://en.wikipedia.org/wiki/Continuous-time_Markov_chain#Embedded_Markov_chain . What you have looks odd. – A.S. Jan 25 '16 at 15:56
  • As I said, there are two possible discretizations of a continuous Markov process $(X_t){t\in\mathbb R+}$ that can be called its skeleton. Either one fixes a times step $\delta$ and one defines $(\xi_n){n\in\mathbb N}$ by $\xi_n=X{n\delta}$. Or one considers the sequence $(\eta_n){n\in\mathbb N}$ of different states occupied by $(X_t)$, that is, one defines recursively $\tau_0=0$ and $\tau{n+1}=\inf{t>\tau_n\mid X_t\ne X_{\tau_n}}$, and one sets $\eta_n=X_{\tau_n}$. Your second matrix works neither for $(\xi_n)$ nor for $(\eta_n)$. – Did Jan 25 '16 at 19:21

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