Temporal intensity and likelihood

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.

Fix a supplied event history $h$ at time $t$ in the model of Temporal point-process models and conditional intensity. Let ${\lambda }_h^{\ast }(u)$ be its nonnegative no-event conditional hazard, supplied as a locally integrable version on every finite interval, and put ${\mathrm{\Lambda }}_h(u):={\int }_t^u{\lambda }_h^{\ast }(s)ds\mathrm{.}$ For each finite $U>t$, on $t\le u\le U$ the conditional survivor and finite- time density of the next event are $S_h(u)=\exp (-{\mathrm{\Lambda }}_h(u)),f_h(u)={\lambda }_h^{\ast }(u)\exp (-{\mathrm{\Lambda }}_h(u))\text{for almost every }u\mathrm{.}$ If ${\mathrm{\Lambda }}_h(u)\to {\mathrm{\Lambda }}_h(\mathrm{\infty })<\mathrm{\infty }$ as $u\to \mathrm{\infty }$, then $K_h(\{\mathrm{\infty }\})=\exp (-{\mathrm{\Lambda }}_h(\mathrm{\infty }))$. If ${\mathrm{\Lambda }}_h(u)\to \mathrm{\infty }$, this terminal mass is zero. These conclusions concern one next-event law and do not prove non-explosion of an entire process.

Rasmussen proves the hazard-to-survival relation by integrating the logarithmic derivative of the survivor, then differentiates to obtain the density. Kass, Eden, and Brown give the same conditional hazard identity and derive the waiting-time density in their full Section 19.4. The proof above keeps the finite-cumulative-hazard case, including its possible mass at $+\mathrm{\infty }$, separate from process existence and non-explosion.

Let $T>0$, $\theta >0$, and let $S=(0,T]$ carry its subspace topology and Borel sigma-algebra. This space is Hausdorff as a subspace of $\mathbb{R}$; rational intervals give a countable base; and each $x\in S$ has a compact neighborhood $[x-\delta ,x+\delta ]\cap S$ for some $0<\delta <x$. Let $\mathrm{\ell }$ be Lebesgue measure restricted to $S$. There exists a Poisson point process on $S$ with intensity measure ${\nu }_{\theta }(B)=\theta \mathrm{\ell }(B)$. Its total count is finite almost surely, its event times have no ties almost surely, and for every integer $m\ge 0$ the joint density of the ordered event times $0<t_1<\cdots <t_m\le T,N(S)=m,$ with respect to $m$-dimensional Lebesgue measure on the ordered simplex is $e^{-\theta T}{\theta }^m\mathrm{.}$ For $m>0$, conditional on $N(S)=m$, the ordered event times have density $m!\mathrm{/}{T}^m$. For $m=0$ the empty-pattern probability is $e^{-\theta T}$.

Rasmussen derives the homogeneous Poisson event-time likelihood as a special case of the conditional-intensity formula. Kass, Eden, and Brown derive the ordered event-time density from independent Poisson increments. The local construction and diagonal-null argument provide a finite-window existence and simplicity proof.

Assume AC, Countable Choice and Dependent Choice for the cited likelihood and hazard-calculus interfaces. Fix $T>0$ and a common specified initial history. Let $(P_{\theta })$ be a supplied family of laws for actual simple, non-explosive temporal point-process models on $(0,T]$ satisfying Temporal point-process models and conditional intensity, with jointly measurable conditional history-update and waiting-time kernels. Let $P_0$ be the supplied unit-rate Poisson reference on the finite-pattern space ${\mathsf{X}}_T={⨆}_{m\ge 0}{\mathrm{\Delta }}_m,{\mathrm{\Delta }}_0=\{\mathrm{\varnothing }\},{\mathrm{\Delta }}_m=\{(t_1,\dots ,t_m):0<t_1<\cdots <t_m\le T\},$ equipped with the disjoint-union Borel sigma-algebra. For a pattern $0<t_1<\cdots <t_m\le T$, let $h_{i-1}$ be the initial history updated by $t_1,\dots ,t_{i-1}$. On each simplex, the laws are dominated by $P_0$ and the following chosen density representative gives the Radon–Nikodym derivative $P_0$-almost everywhere: $\frac{d{P}_{\theta }}{d{P}_0}(t_1,\dots ,t_m)=e^T({\prod }_{i=1}^m{\lambda }_{\theta }^{\ast }(t_i\mid h_{i-1}))\exp \text{\hspace{0.17em}⁣}(-{\int }_0^T{\lambda }_{\theta }^{\ast }(u\mid h_{u-})du)\mathrm{.}$ Here $h_{u-}$ is the history immediately before $u$. Thus the likelihood as a function of $\theta$ is proportional to the product of pre-event intensities times the terminal integrated-hazard factor; $e^T$ is parameter-independent. If an observed event has zero intensity, the density is zero; a finite log-likelihood requires positive intensity at each observed event. For $m=0$, the event product is the empty product $1$ and the terminal no-event factor remains.

For marked patterns, let marks lie in a supplied measurable space with a common mark probability measure $Q$. Here $h_i$ records all past event times and marks, and the marked-history update is supplied and measurable. Require that the supplied waiting-time kernels, conditioned on this full marked history, satisfy the same temporal model; an unmarked marginal waiting-time law alone is not this hypothesis. If each event also has a supplied jointly measurable conditional mark density $q_{\theta }(k_i\mid t_i,h_{i-1})$ relative to $Q$, and the reference uses independent marks with law $Q$, evaluate the displayed intensity factors on these marked histories and multiply the ratio by ${\prod }_i{q}_{\theta }(k_i\mid t_i,h_{i-1})$. The ground intensity does not determine a mark law.

Rasmussen's Proposition 3.1 gives the finite-window likelihood and derives it by multiplying inter-event densities with the terminal no-event probability; its marked case includes conditional mark-density factors. Kass, Eden, and Brown independently derive the same factorization in Section 19.4. This theorem is conditional on supplied process laws and makes no existence or stationarity claim from an intensity formula.