 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 satisfyingIncrements over disjoint time intervals are independent. In particular,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 $1-e^{-\lambda h}\to 0$.