The Compensated Poisson Process is an Ito Integrator with Its Intensity

Let $\lambda_{\mathrm{int}}:[0,\infty)\to\mathbb{R}$ be an intensity function with mean function $\Lambda$ in the sense of Stochastic Process, Independent Increments, and Inhomogeneous Poisson Process (intensity functions are nonnegative, so $\lambda_{\mathrm{int}}$ may be regarded as a function into $[0,\infty)$, as required of an intensity in Ito Integrator of Intensity Type), let $N=(N_t)_{t\ge0}$ be an inhomogeneous Poisson process with intensity $\lambda_{\mathrm{int}}$ on a probability space $(\Omega,\mathcal{F},P)$, and let $M=(M_t)_{t\ge0}$ with $M_t=N_t-\Lambda(t)$ be the compensated Poisson process of The Compensated Poisson Process is a Square-Integrable Martingale. Let $(\mathcal{F}^{N}_t)_{t\ge0}$ be the natural filtration of $N$.

Then the pair $(M,\lambda_{\mathrm{int}})$ is an It^{o} integrator of intensity type with respect to $(\mathcal{F}^{N}_t)_{t\ge0}$. Moreover, the natural filtrations of $M$ and of $N$ coincide: $\mathcal{F}^{M}_t=\mathcal{F}^{N}_t$ for every $t\ge0$.