N-Agent Driving System

Let $N$, $l$, and $\tilde{l}$ be natural numbers with $N\ge 1$, $l\ge 2$, and $\tilde{l}\ge 1$.

An $N$-agent driving system with $l$ states and $\tilde{l}$ observation channels is a probability space $(\Omega,\mathcal{F},P)$ together with the following data.

1. (Initial states.) Random variables $\varsigma^1_0,\dots,\varsigma^N_0$ on $(\Omega,\mathcal{F},P)$, each taking values in $\{1,\dots,l\}$.

2. (Transition clocks.) For each $i\in\{1,\dots,N\}$ and each ordered pair $(\sigma,\gamma)$ with $\sigma,\gamma\in\{1,\dots,l\}$ and $\sigma\neq\gamma$, a stochastic process $Y^{i,\sigma\gamma}=(Y^{i,\sigma\gamma}_u)_{u\ge0}$ on $(\Omega,\mathcal{F},P)$ that is a homogeneous Poisson process with rate $1$ and such that for every $\omega\in\Omega$ the path $u\mapsto Y^{i,\sigma\gamma}_u(\omega)$ is a counting path.

3. (Observation clocks.) For each $i\in\{1,\dots,N\}$ and each $\upsilon\in\{1,\dots,\tilde{l}\}$, a stochastic process $\tilde{Y}^{i,\upsilon}=(\tilde{Y}^{i,\upsilon}_u)_{u\ge0}$ on $(\Omega,\mathcal{F},P)$ that is a homogeneous Poisson process with rate $1$ and all of whose paths are counting paths.

4. (Independence.) The following finite family of $\sigma$-algebras is independent: the $\sigma$-algebra generated by the initial states $\varsigma^1_0,\dots,\varsigma^N_0$; for each transition clock, the $\sigma$-algebra $\sigma(Y^{i,\sigma\gamma}_u:u\ge0)$ generated by its variables; and, for each observation clock, the $\sigma$-algebra $\sigma(\tilde{Y}^{i,\upsilon}_u:u\ge0)$.

The processes $Y^{i,\sigma\gamma}$ are called transition clocks and the processes $\tilde{Y}^{i,\upsilon}$ are called observation clocks.