= Solution
A homogeneous <Poisson process> of rate $\lambda>0$ is an integer-valued <counting process> with $N_0=0$ almost surely, nondecreasing right-continuous paths with unit jumps, and <independent> <stationary increments> satisfying
$$
\boxed{N_t-N_s\sim\operatorname{Poisson}(\lambda(t-s)),\qquad0\leq s<t.}
$$
Increments over disjoint time <intervals> are <independent>. In particular,
$$
\mathbb P(N_t=k)=e^{-\lambda t}\frac{(\lambda t)^k}{k!},\qquad k=0,1,2,\ldots.
$$
The finite Poisson counts ensure that there is no explosion in finite time. The increment formula also gives <stochastic continuity>, since the <probability> of any arrival in an <interval> of length $h$ is $1-e^{-\lambda h}\to0$.
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