The Compensated Poisson Process is a Square-Integrable Martingale

Let $\lambda:[0,\infty)\to\mathbb{R}$ be an intensity function with mean function $\Lambda$ in the sense of Stochastic Process, Independent Increments, and Inhomogeneous Poisson Process, where $\mathbb{R}$ is the set of real numbers, and let $N=(N_t)_{t\ge0}$ be an inhomogeneous Poisson process with intensity $\lambda$ on a probability space $(\Omega,\mathcal{F},P)$. Let $(\mathcal{F}^{N}_t)_{t\ge0}$ be the natural filtration of $N$.

Then the compensated Poisson process $M=(M_t)_{t\ge0}$ defined by

is a square-integrable martingale with respect to $(\mathcal{F}^{N}_t)_{t\ge0}$.