Markov Jump Processes

We want to describe Markov processes that evolve through continuous time t ≥ 0, but in a discrete state space ℒ. The prescription for such a process has two ingredients. There are random jump times 0 < τ₁ < τ₂ < … < τ_n < … when the process jumps away from the state it is at, and there are transition probabilities Q _xy that govern the transitions at these jump times. The process {X(t): t ≥ 0 } itself has piecewise constant paths, which we can take to be right-continuous $$X(t) = \left\{ \begin{gathered} {x_0}, 0 \leqslant t < {\tau _1}, \hfill \\ {x_1}, {\tau _1} \leqslant t < {\tau _2}, \hfill \\ {x_2}, {\tau _2} \leqslant t < {\tau _3}, \hfill \\ \ldots \ldots \ldots \ldots \ldots \ldots \ldots \hfill \\ \end{gathered} \right.$$