Transition Labels and Aggregate Transition Clocks

Let $l\ge2$ be a natural number. A transition label on $l$ states is an ordered pair $c=(\sigma,\gamma)$ of elements of $\{1,\dots,l\}$ with $\sigma\neq\gamma$; the transition labels form the index set of a transition-rate family on $l$ states.

Let $(\Omega,\mathcal{F},P)$ be a probability space. A family of aggregate transition clocks on $l$ states over $(\Omega,\mathcal{F},P)$ is a family $\mathsf{P}=(\mathsf{P}^{c})$, indexed by the transition labels $c$ on $l$ states, of stochastic processes $\mathsf{P}^{c}=(\mathsf{P}^{c}_u)_{u\ge0}$ on $(\Omega,\mathcal{F},P)$ such that:

1. each $\mathsf{P}^{c}$ is a homogeneous Poisson process with rate $1$;

2. for every $c$ and every $\omega\in\Omega$ the path $u\mapsto\mathsf{P}^{c}_u(\omega)$ is a counting path;

3. the family of $\sigma$-algebras $\sigma(\mathsf{P}^{c}_u:u\ge0)$ generated by the individual clocks, indexed by the transition labels, is independent.

For $\omega\in\Omega$ we write $\mathsf{P}(\omega)$ for the family of counting paths $(u\mapsto\mathsf{P}^{c}_u(\omega))$ indexed by the transition labels.