Finite-state generator chain

Let $E=\{1,\dots ,m\}$ for a finite integer $m\ge 1$. A rate matrix on $E$ is a real matrix $L=(L_{ij}{)}_{i,j\in E}$ such that $L_{ij}\ge 0$ for $i\mathrm{\ne }j$ and ${\sum }_{j\in E}{L}_{ij}=0$ for every $i$. Write ${\lambda }_i={\sum }_{j\mathrm{\ne }i}{L}_{ij}$, so $L_{ii}=-{\lambda }_i$. The row convention acts on a column observable $f:E\to \mathbb{R}$ by $(Lf)(i)={\sum }_j{L}_{ij}f(j)$; a row probability vector evolves by right multiplication.

Fix an initial probability vector $\nu =({\nu }_i{)}_{i\in E}$, meaning ${\nu }_i\ge 0$ and ${\sum }_{i\in E}{\nu }_i=1$; initial law $\nu$ means $\mathbb{P}(Y_0=i)={\nu }_i$ for every $i\in E$. Put $q={\max }_i{\lambda }_i$. If $q=0$, then $L=0$ and set $N_t=0$ for every $t\ge 0$ and let the associated realization be the constant process $X_t=Y_0$ with $Y_0\sim \nu$. If $q>0$, put $R=I+L\mathrm{/}q$, a stochastic matrix. A uniformized realization consists of a probability space carrying a discrete-time Markov sequence $(Y_n{)}_{n\ge 0}$ with initial law $\nu$ and transition matrix $R$, and independent random variables $(W_n{)}_{n\ge 1}$ that are independent of the entire sequence $Y$ and satisfy $\mathbb{P}(W_n>u)=e^{-qu}$ for $u\ge 0$. Here a discrete-time Markov sequence with transition matrix $R$ means that $\mathbb{P}(A\cap \{Y_{n+1}=j\})=\mathbb{E}[{\mathbf{1}}_A{R}_{Y_{n}j}]$ for every $n\ge 0$, $j\in E$, and $A\in \sigma (Y_0,\dots ,Y_n)$. Each $W_n$ therefore has the exponential distribution of rate $q$; in particular its mean is $q^{-1}$. Require ${\sum }_{n\ge 1}{W}_n=+\mathrm{\infty }$ almost surely. Put $T_0=0$, $T_n={\sum }_{k=1}^n{W}_k$, and, on this probability-one event, $N_t=\max \{n:T_n\le t\}$ and $X_t=Y_{N_t}$. On its null complement set $N_t=0$ and $X_t=Y_0$. The construction filtration is ${\mathcal{F}}_t=\sigma (N_s,X_s:0\le s\le t)$; it records event counts and states through time $t$ and is not completed or made right-continuous by this definition.

This uniformized construction permits reducible rate matrices and states with zero exit rate. Under the Axiom of Choice, Uniformization constructs every finite-state rate matrix chain ↗ proves that every finite rate matrix and initial law admit such a realization and that its conditional transition matrices are $e^{tL}:={\sum }_{n=0}^{\mathrm{\infty }}{t}^n{L}^n\mathrm{/}n!$.

Uniformization constructs every finite-state rate matrix chain ↗