Markov Process and a continuous PDF

Question

I read an economics paper, and I got quite confused about the setting of the model: they assume that the labor productivity $\varepsilon$ (defined on a discrete state space) follows a Markov process. In the simulation/calibration part, they specify their assumption further: the labor productivity is drawn from a bounded Pareto distribution with a fixed probability $\delta_{\varepsilon}$ that the labor productivity will get redrawn.

I know that there are three key elements for a discrete Markov process (from textbook definition): state space $S=\{\varepsilon_{1}, ..., \varepsilon_{n}\}$, transition matrix $P_{n\times n}$, and the initial probability $\pi_{0}=\begin{bmatrix} P(X_{0}=\varepsilon_{1})\\ \vdots\\ P(X_{0}=\varepsilon_{n}) \end{bmatrix}$, where $X_{0}$ is drawn from some discrete distribution.

My questions are:

1. How could they draw the labor productivity from a continuous distribution (i.e., bounded Pareto distribution)?
2. How could they specify the transition matrix $P_{n\times n}$ and the initial probability $\pi_{0}$ in this case if allows assuming a fixed probability to get the labor productivity redrawn?

Answer 1

There is also theory about Markov Processes defined on continuous state spaces, you can google for that. Your definitions are for discrete state spaces, where everything's a little bit more straightforward.