Brownian Motion is an Ito Integrator with Unit Intensity

Let $B=(B_t)_{t\ge0}$ be a standard Brownian motion on a probability space $(\Omega,\mathcal{F},P)$, and let $(\mathcal{F}^{B}_t)_{t\ge0}$ be its natural filtration.

Then $B$ is a square-integrable martingale with respect to $(\mathcal{F}^{B}_t)_{t\ge0}$, and the pair $(B,\rho)$ with constant intensity $\rho\equiv1$ is an It^{o} integrator of intensity type with respect to $(\mathcal{F}^{B}_t)_{t\ge0}$.