I´m trying to prove that a Markov-Modulated Poisson process could be seen as a 2-dimensional continuous time markov chain.
For this, I'm considering a two state markov chain $\{J(t) : t \geq 0\}$ with Q-matrix $$\begin{pmatrix}-\alpha_1 & \alpha_1\\ \alpha_2 &-\alpha_2 \end{pmatrix}$$ and two positive numbers $\beta_1$ and $\beta_2$.
Thereby, if $\{N(t) : t\geq 0\}$ is a cox-proccess with rate $\lambda(t) = \beta_{J(t)}$, then $\{(N(t),J(t)) : t \geq 0\}$ is a two state markov chain.
I supuse that this result is true. Moreover, I think that $\{N(t) : t \geq 0\}$ is a two state markov chain with state space $E = \mathbb{N}_0 \times\{1,2\}$,* jump matrix given by $$\Pi_{(i,j),(k,l)} = \begin{cases} \frac{\beta_{j}}{\alpha_j + \beta_{j}},& \text{ si } k = i+1 \text{ y } j = l. \\ \frac{\alpha_j}{\alpha_j + \beta_{j}}, & \text{ si }k= i \text{ y } l \in\{1,2\}\setminus\{j\}.\\ 0, &\text{e.o.c}. \end{cases}$$ and $Q$-matrix given by $$ Q_{(i,j),(k,l)} = \begin{cases} \beta_{j},& \text{ si } k = i+1 \text{ y } j = l. \\ \alpha_j, & \text{ si }k= i \text{ y } l \in\{1,2\}\setminus\{j\}.\\ -(\alpha_j + \beta_{j}), & \text{ si }k= i \text{ y } l = j.\\ 0, &\text{e.o.c}. \end{cases}$$ I've tried to prove this by using the arrival times of the cox process and the markov chain $J(t)$, but I havent achieve anithing up to now. So, I'll be very glad with any sugestion.
*Where $\mathbb{N}_0 = \{0,1,2,3,...\}$.