Survival Identity, Measurability, Event-Count Recursion, and Normalization of the Likelihood of a Causal Intensity

Let $\tilde{l}\ge1$ be a natural number, let $T>0$ be a real number, let $(\mathbf{R},\mathcal{R},\rho)$ be the observation record space with horizon $T$ and $\tilde{l}$ channels, with channel set $V$ and cells $C_\emptyset$ and $C_{k,v}$ as there, and let $\lambda=(\lambda^\upsilon)_{\upsilon\in V}$ be a causal intensity on $\mathbf{R}$ with bound $\bar\lambda$, total intensity $\lambda^{\mathrm{tot}}$, and likelihood $\ell_\lambda$. Integrals over subsets of $[0,T]$ are Lebesgue integrals with respect to the restricted Lebesgue measure on $[0,T]$, written $\int_{[a,b]}h(u)\,du=\int_{[0,T]}\mathbf{1}_{[a,b]}(u)h(u)\,du$ and likewise for $(a,b]$, with $\mathbf{1}_A$ the indicator of $A$; $\exp$ is the real exponential function, $k!$ the factorial with $0!=1$, finite products over an empty index range and zeroth powers such as $\bar\lambda^{0}$ are $1$, and integrals of $[0,\infty]$-valued measurable functions on $\mathbf{R}$ are those of Lebesgue Integral of a Nonnegative Measurable Function. For $k\ge0$ let $\mathbf{R}^{(k)}\subseteq\mathbf{R}$ be the set of records with exactly $k$ events, the union of the cells $C_{k,v}$ over $v\in V^k$ (so $\mathbf{R}^{(0)}=C_\emptyset$), and write $\int_{\mathbf{R}^{(k)}}h\,d\rho=\int_{\mathbf{R}}h\mathbf{1}_{\mathbf{R}^{(k)}}\,d\rho$.

For $r=(k,t,v)\in\mathbf{R}$ and $s\in[0,T]$ put $\Lambda_s(r)=\int_{[0,s]}\lambda^{\mathrm{tot}}_u(r)\,du,\qquad q(r)=\Bigl(\prod_{i=1}^{k}\lambda^{v_i}_{t_i}(r)\Bigr)\exp\bigl(-\Lambda_{t_k}(r)\bigr),$ with the convention $t_0=0$ when $k=0$, so that $q(r_\emptyset)=1$; we call $q$ the pre-survival density of $\lambda$. Put $Q_n=\int_{\mathbf{R}^{(n)}}q\,d\rho$ for $n\ge0$.

1. (Survival identity) Let $\bar\mu\ge0$ and let $\mu:[0,T]\to[0,\bar\mu]$ be measurable with respect to the trace Borel $\sigma$-algebra, and put $M_s=\int_{[0,s]}\mu(u)\,du$ for $s\in[0,T]$. Then $s\mapsto M_s$ is continuous on $[0,T]$ (for the metric of the real line), and for all real $0\le a\le b\le T$, $\int_{(a,b]}\mu(u)\exp(-M_u)\,du=\exp(-M_a)-\exp(-M_b).$

2. (Measurability and bounds) For every $s\in[0,T]$ the map $r\mapsto\Lambda_s(r)$ is $\mathcal{R}$-measurable, and the functions $\ell_\lambda$ and $q$ are $\mathcal{R}$-measurable; for $r\in\mathbf{R}^{(k)}$, $0\le\ell_\lambda(r)\le q(r)\le\bar\lambda^{k}$ and $\ell_\lambda(r)=q(r)\exp\bigl(-(\Lambda_T(r)-\Lambda_{t_k}(r))\bigr)$; and $Q_n\le\frac{(\tilde{l}\bar\lambda T)^{n}}{n!}\qquad(n\ge0),\qquad Q_0=1 .$

3. (Event-count recursion) For every $n\ge0$, $Q_n=\int_{\mathbf{R}^{(n)}}\ell_\lambda\,d\rho+Q_{n+1}.$

4. (Normalization) For every $n\ge0$, $\sum_{k=0}^{n}\int_{\mathbf{R}^{(k)}}\ell_\lambda\,d\rho=1-Q_{n+1}$, and $\int_{\mathbf{R}}\ell_\lambda\,d\rho=1 .$