Stochastic Process, Independent Increments, and Inhomogeneous Poisson Process

Let $(\Omega,\mathcal{F},P)$ be a probability space and let $\mathbb{R}$ be the real numbers.

A stochastic process on $[0,\infty)$ is a family $X=(X_t)_{t\ge0}$ of random variables on $(\Omega,\mathcal{F},P)$ indexed by the nonnegative real numbers. The process $X$ has independent increments if for all real $0\le t_0<t_1<\dots<t_r$ the random variables $X_{t_1}-X_{t_0},\dots,X_{t_r}-X_{t_{r-1}}$ are independent. (Differences of random variables are random variables: for every real $u$, $\{X_t-X_s>u\}=\bigcup_{q\in\mathbb{Q}}(\{X_t>q\}\cap\{X_s<q-u\})$, where $\mathbb{Q}$ denotes the rational numbers, using claim 1 of density of the rationals and the generator criterion of Measurable Map, and the Set of Measurable Maps.)

An intensity function is a nonnegative function $\lambda:[0,\infty)\to\mathbb{R}$ that is continuous at every point of $[0,\infty)$ relative to $[0,\infty)$, both $[0,\infty)$ and the codomain $\mathbb{R}$ carrying the metric of the real line. Its mean function is

$$\Lambda(t)=\int_0^t\lambda(s)\,ds\qquad(t\ge0),$$

the Riemann integral, which for $t>0$ exists because the restriction $\lambda|_{[0,t]}$ is continuous by claim 1 of Restriction Stability of Continuity and of the Derivative, so that claim 3 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval applies, with $\Lambda(0)=0$. By additivity on adjacent intervals (Additivity of the Riemann Integral over Adjacent Intervals, Subintervals and Oriented Integrals, and Integrals of Restrictions), $\Lambda(t)-\Lambda(s)=\int_s^t\lambda(u)\,du$ for $0\le s\le t$, that lemma also yielding integrability of the restrictions, and this is nonnegative because every lower sum of a nonnegative function is nonnegative (upper and lower sums); hence $\Lambda$ is nondecreasing.

A stochastic process $N=(N_t)_{t\ge0}$ on $(\Omega,\mathcal{F},P)$ is an inhomogeneous Poisson process with intensity $\lambda$ if:

1. $N_0=0$;

2. $N$ has independent increments;

3. for all $0\le s<t$, the increment $N_t-N_s$ has the Poisson distribution with parameter $\Lambda(t)-\Lambda(s)$, in the sense of Distribution Functions.

If $\lambda$ is constant with value $\theta\ge0$, then $\Lambda(t)=\theta t$ (the Riemann integral of a constant, directly from the upper and lower sums), and $N$ is called a homogeneous Poisson process with rate $\theta$.