Kolmogorov Forward Equations for the Inhomogeneous Poisson Process

Let $\lambda$ be an intensity function with mean function $\Lambda$, and let $N=(N_t)_{t\ge0}$ be an inhomogeneous Poisson process with intensity $\lambda$ on a probability space $(\Omega,\mathcal{F},P)$. Here $\mathbb{N}$ denotes the set of natural numbers, $\mathbb{N}_0=\mathbb{N}\cup\{0\}$ the set of nonnegative integers, and $\mathbb{R}$ the set of real numbers. For $k\in\mathbb{N}_0$ define

$$p_k(t)=P(N_t=k)\qquad(t\ge0).$$

Then the following hold.

Claim 1 (explicit form). For every $k\in\mathbb{N}_0$ and $t\ge0$,

$$p_k(t)=\exp(-\Lambda(t))\,\frac{\Lambda(t)^{k}}{k!},$$

with the exponential function and the factorial, under the conventions $0!=1$ and $\Lambda(t)^{0}=1$ recorded in Poisson Distribution.

Claim 2 (forward equations). Each $p_k$ has a derivative at every $t>0$, and a one-sided derivative at $t=0$ given by the same limit restricted to positive increments; with these derivatives the Kolmogorov forward equations hold for all $t\ge0$:

$$p_0'(t)=-\lambda(t)\,p_0(t),\qquad p_k'(t)=\lambda(t)\,p_{k-1}(t)-\lambda(t)\,p_k(t)\quad(k\ge1),$$

with the initial values $p_0(0)=1$ and $p_k(0)=0$ for $k\ge1$.

Claim 3 (uniqueness). If $(q_k)_{k\in\mathbb{N}_0}$ is any family of functions $q_k:[0,\infty)\to\mathbb{R}$, differentiable in the same sense, satisfying the same system of equations and the same initial values, then $q_k=p_k$ for every $k\in\mathbb{N}_0$. In particular the functions $t\mapsto P(N_t=k)$ form the unique solution of this system of ordinary differential equations.