Temporal point-process models and conditional intensity

Assume the Axiom of Choice as in Simple counting-process intensity and compensator contract. Fix a finite horizon $T>0$, a probability space with an increasing filtration $({\mathcal{F}}_t)$ in the all-pairs convention of Continuous-time filtrations and all-pairs martingales, and an adapted càdlàg (right-continuous with left limits), nondecreasing, integer-valued process $N$ with $N_0=0$, $N_T<\mathrm{\infty }$ almost surely, and jumps $\mathrm{\Delta }{N}_t\in \{0,1\}$ outside one null event. Such a process is simple: its event times are strictly ordered, $0<T_1<\cdots <T_{N_T}\le T$. It is locally finite when it has finitely many events on every finite interval.

For conditional-law calculations, the model supplies a measurable history space $(\mathsf{H},\mathcal{G})$, an adapted measurable history random element $H_t$ recording the initial information and events observed through time $t$, and a measurable update map $h\mapsto h\oplus u$ that appends an event at time $u$. Write ${\mathcal{H}}_t:=\sigma (H_t)$ and let $H_{t-}$ record only events strictly before $t$. Relevant exogenous covariates are included in the supplied history. Measurability has the meaning of A measurable function between measurable spaces. Predictability is measurability for the predictable sigma-algebra generated by $(s,u]\times A$ with $A\in {\mathcal{F}}_s$ and the time-zero generators, as in Progressively measurable and predictable processes.

The model supplies a probability kernel $K_{t,h}$ on $(t,\mathrm{\infty }]$ for the next event time given history $h$ at time $t$ (Measure kernel and probability kernel, Conditional law given a random element); its finite-time density versions are jointly measurable in $(t,h,u)$. The value $+\mathrm{\infty }$ means that no later event occurs. If the finite-time measure is absolutely continuous with respect to Lebesgue measure (Absolute continuity of a signed or complex measure with respect to a positive measure, Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$, Assuming countable choice, every Borel subset of ${\mathbb{R}}^n$ is Lebesgue measurable), let $f_{t,h}$ be its supplied nonnegative Radon--Nikodym density (The Radon-Nikodym derivative as an almost-everywhere equivalence class). Write $S_{t,h}(u):=K_{t,h}((u,\mathrm{\infty }])$ for its survivor and define its no-event conditional hazard by ${\lambda }_{t,h}^{\ast }(u):=\frac{f_{t,h}(u)}{S_{t,h}(u)}\text{where }{S}_{t,h}(u)>0.$ Assume these hazards are locally integrable on every finite interval. The model also supplies a nonnegative predictable effective-intensity process ${\lambda }^{\ast }$ compatible with these waiting-time hazards: for each history $h$ at time $t$, on its supplied measurable no-event continuation up to $u>t$, the pre-event intensity ${\lambda }^{\ast }(u\mid h_{u-})$ agrees with ${\lambda }_{t,h}^{\ast }(u)$ for Lebesgue-almost every $u$. The jointly measurable hazard representatives used in the event-density factors are these same versions on their proper domains $u>t$. Compatibility with the predictable process is a model requirement, not an evaluation of a waiting-time density at its excluded left endpoint. This pre-event convention means the event at $u$ does not change its own intensity.

The temporal conditional-intensity model requires both this hazard-compatible conditional law and the separate counting-process compensator contract of Simple counting-process intensity and compensator contract: if an at-risk indicator $Y$ and intensity density $\lambda$ are separately specified, the effective intensity is ${\lambda }^{\ast }=Y\lambda$; without an at-risk indicator, it is ${\lambda }^{\ast }$. Then $A_t={\int }_0^t{\lambda }^{\ast }(u)du$ is finite pathwise, continuous, adapted and predictable, $\mathbb{E}[A_T]<\mathrm{\infty }$, and $N-A$ is a square-integrable martingale in the all-pairs convention of Continuous-time adapted processes and martingales. At-risk exposure, if present, is part of the effective intensity. The compensator clause is a model assumption: a displayed hazard formula alone proves neither that a process with this law exists nor that the martingale contract holds. No stationarity or non-explosion claim follows from writing an intensity specification.

Rasmussen defines conditional intensity through the conditional waiting-time hazard and makes the terminating-process case explicit. Kass, Eden, and Brown use the pre-event history convention and distinguish an intensity formula from the assumptions needed for a point-process law. The local compensator contract is inherited exactly from the counting-process page.