Conditional-intensity inter-event law and point-pattern likelihood

For $m\ge 1$, recursively define the density of the first $i$ event times by $g_0=1$ and $g_i(t_1,\dots ,t_i)=g_{i-1}(t_1,\dots ,t_{i-1})f_{\theta }(t_i\mid h_{i-1})$ on the ordered prefix simplex. At $i=1$ this is the supplied conditional density. If the first $i-1$ times have density $g_{i-1}$, conditioning on that prefix and integrating its supplied probability kernel for the next waiting time multiplies the density by $f_{\theta }(t_i\mid h_{i-1})$; iterated kernel integration proves the formula for $i$. The measurable update and kernel assumptions in [F1] make each factor measurable. For exactly $m$ events by $T$, the final history must have no event in $(t_m,T]$, whose conditional probability is $S_{\theta }(T\mid h_m)$. Thus the pattern density is ${\prod }_{i=1}^m{f}_{\theta }(t_i\mid h_{i-1})S_{\theta }(T\mid h_m)$; for $m=0$ it is the initial no-event probability. The simplices exhaust the pattern space because the event count is finite almost surely.

By [F2], each event-time density is its pre-event intensity times $\exp \text{\hspace{0.17em}⁣}(-{\int }_{t_{i-1}}^{t_i}{\lambda }_{\theta }^{\ast }(u\mid h_{u-})du)$, where $t_0=0$, and the terminal survivor is $\exp (-{\int }_{t_m}^T{\lambda }_{\theta }^{\ast }(u\mid h_{u-})du)$. Multiplying these terms adds the integrals over the adjacent intervals $(0,t_1],\dots ,(t_m,T]$, so on ${\mathrm{\Delta }}_m$ the pattern density is $p_{\theta }(t_1,\dots ,t_m)=({\prod }_{i=1}^m{\lambda }_{\theta }^{\ast }(t_i\mid h_{i-1}))\exp \text{\hspace{0.17em}⁣}(-{\int }_0^T{\lambda }_{\theta }^{\ast }(u\mid h_{u-})du)$. For $m=0$ there are no event factors and the same expression uses the empty product.

By [F3], the reference density equals $e^{-T}>0$ on each ${\mathrm{\Delta }}_m$ and its empty-pattern mass is positive. If a measurable set in ${\mathsf{X}}_T$ has $P_0$-measure zero, its section in every ${\mathrm{\Delta }}_m$ is Lebesgue-null and it excludes the empty pattern. Step 2.1 gives a Lebesgue density for $P_{\theta }$ on every simplex, so each section has $P_{\theta }$-measure zero; countable additivity over the disjoint union gives $P_{\theta }\ll P_0$. By [F4], the Radon–Nikodym derivative is the ratio of the two pattern densities, namely $p_{\theta }\mathrm{/}{e}^{-T}=e^T{p}_{\theta }$, $P_0$-almost everywhere. If any factor at an observed event is zero, this representative is zero; when all are positive, taking logarithms and discarding the parameter-free term $T$ gives the stated likelihood form.

In the marked case, condition successively on the full past marked history: the next event time has its supplied waiting-time kernel under that history, and then the new mark has its supplied conditional density $q_{\theta }(k_i\mid t_i,h_{i-1})$. Update the history by both time and mark before taking the next waiting-time kernel. The measurable updates and density hypotheses allow the same iterated integration as in step 1.1, so the joint pattern measure acquires one mark-density factor per event in addition to the waiting-time and terminal-survivor factors evaluated on marked histories. Relative to the reference product mark measure $Q^{\otimes m}$, whose density is $1$, these factors multiply the unmarked density ratio from step 3.1. This uses the supplied mark law and introduces no mark distribution assumption from the ground intensity.