Simple counting-process intensity and compensator contract

Assume the Axiom of Choice The Axiom of Choice. Fix a finite horizon $T>0$, a probability space $(\mathrm{\Omega },\mathcal{F},P)$ and a continuous-time filtration $({\mathcal{F}}_t{)}_{0\le t\le T}$ in the all-pairs convention of Continuous-time filtrations and all-pairs martingales. A simple unit-jump counting process is an adapted, càdlàg (right-continuous with left limits), nondecreasing, integer-valued process $N$ with $N_0=0$, $N_T<\mathrm{\infty }$ almost surely, and $\mathrm{\Delta }{N}_t\in \{0,1\}$ for every $0<t\le T$ outside one null event; here $\mathrm{\Delta }{N}_t:=N_t-N_{t-}$.

An at-risk process is a predictable process $Y:[0,T]\times \mathrm{\Omega }\to \{0,1\}$, and an intensity density is a finite, nonnegative predictable process $\lambda$. Form the Lebesgue integral $A_t:={\int }_0^t{Y}_s{\lambda }_{s}ds,0\le t\le T\mathrm{.}$ For this model and every nonnegative predictable process $G$, define the pathwise energy integral by ${\int }_0^t{G}_{s}d{A}_s:={\int }_0^t{G}_s{Y}_s{\lambda }_{s}ds,t\le T\mathrm{.}$ The continuous intensity compensator contract requires the defining integral for $A$ to be finite pathwise, and requires $A$ to be continuous, adapted and predictable with $A_0=0$, and that $\mathbb{E}[A_T]<\mathrm{\infty }$. Set $M_t:=N_t-A_t$. The model contract is the additional hypothesis that $M$ is a square-integrable martingale relative to the stated filtration. Under this contract $A$ is a continuous predictable compensator of $N$.

The displayed intensity density alone does not assert that $N-A$ is a martingale, that such a process exists for a given $N$, or that the contract follows from an empirical observation scheme. No general compensator-existence theorem is included here. The Axiom of Choice is inherited from the continuous-time martingale and conditional-expectation interfaces; no further choice is used to state the model contract.

Tian's Unit 8 sets up the one-event process, at-risk indicator and integrated hazard; Andersen–Borgan–Gill–Keiding define the counting-process compensator as a predictable finite-variation process for which $N-A$ is a local martingale. This library contract strengthens that model clause to a square-integrable martingale and imposes $\mathbb{E}[A_T]<\mathrm{\infty }$ for the isometry developed below. It makes no existence claim from an intensity formula.