Ito Integrator of Intensity Type

Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},P)$ be a filtered probability space and let $\lambda$ denote Lebesgue measure on the real line.

An It^{o} integrator of intensity type with respect to $(\mathcal{F}_t)_{t\ge0}$ is a pair $(M,\rho)$ consisting of a stochastic process $M=(M_t)_{t\ge0}$ and a function $\rho:[0,\infty)\to[0,\infty)$, called the intensity, such that:

(i) $M$ is a square-integrable martingale with respect to $(\mathcal{F}_t)_{t\ge0}$;

(ii) $M_0=0$ almost surely;

(iii) for all real $0\le s<t$, the $\sigma$-algebras $\sigma(M_t-M_s)$ and $\mathcal{F}_s$ are independent;

(iv) $\rho$ is measurable with respect to the Borel $\sigma$-algebra (extended by $0$ to all of $\mathbb{R}$), with $\int_{\mathbb{R}}\mathbf{1}_{(0,T]}\,\rho\,d\lambda<\infty$ for every real $T>0$, and for all real $0\le s<t$ the expectation of the squared increment satisfies

the Lebesgue integral of $\rho$ over $(s,t]$, for which we also write $\int_s^t\rho(r)\,dr$.