Poisson Clock with a Horizon

Let $(\Omega,\mathcal{F},P)$ be a probability space and let $R$ be either a real number with $R>0$ or the symbol $+\infty$; for a real number $u\ge0$ write $u\wedge R$ for the smaller of $u$ and $R$, with $u\wedge(+\infty)=u$.

A Poisson clock with horizon $R$ on $(\Omega,\mathcal{F},P)$ is a stochastic process $Y=(Y_u)_{u\ge0}$ on $(\Omega,\mathcal{F},P)$ such that:

1. for every $\omega\in\Omega$ the path $u\mapsto Y_u(\omega)$ is a counting path;

2. $Y$ has independent increments;

3. for all real $0\le s<t$, the increment $Y_t-Y_s$ has the Poisson distribution with parameter $(t\wedge R)-(s\wedge R)$ (a nonnegative real number), in the sense of Distribution and Cumulative Distribution Function of a Random Variable.