Causal Intensity on the Observation Record Space

Let $\tilde{l}\ge1$ be a natural number, let $T>0$ be a real number, and let $(\mathbf{R},\mathcal{R},\rho)=(\mathbf{R}(T,\tilde{l}),\mathcal{R}(T,\tilde{l}),\rho(T,\tilde{l}))$ be the observation record space with horizon $T$ and $\tilde{l}$ channels, with channel set $V=\{1,\dots,\tilde{l}\}$. Let $\pi_{s-}$ ($s\in[0,T]$) be the strict prefix maps on $\mathbf{R}$. Let $\mathcal{B}_{[0,T]}$ be the trace Borel $\sigma$-algebra on $[0,T]$ and let $\mathcal{B}_{[0,T]}\otimes\mathcal{R}$ be the product $\sigma$-algebra on $[0,T]\times\mathbf{R}$.

A causal intensity on $\mathbf{R}$ with bound $\bar\lambda$, a real number with $\bar\lambda\ge0$, is a family $\lambda=(\lambda^\upsilon)_{\upsilon\in V}$ of maps $\lambda^\upsilon:[0,T]\times\mathbf{R}\to\mathbb{R}$ with values in $[0,\bar\lambda]$, written $\lambda^\upsilon_s(r)=\lambda^\upsilon(s,r)$, such that for every $\upsilon\in V$:

(i) (Measurability) $\lambda^\upsilon$ is measurable with respect to $\mathcal{B}_{[0,T]}\otimes\mathcal{R}$ and the Borel $\sigma$-algebra of the real line;

(ii) (Non-anticipation) $\lambda^\upsilon_s(r)=\lambda^\upsilon_s\bigl(\pi_{s-}(r)\bigr)$ for every $s\in[0,T]$ and every $r\in\mathbf{R}$.

The total intensity of $\lambda$ is the map $\lambda^{\mathrm{tot}}:[0,T]\times\mathbf{R}\to\mathbb{R}$, $\lambda^{\mathrm{tot}}_s(r)=\sum_{\upsilon\in V}\lambda^\upsilon_s(r)$, the finite sum of the channel intensities; it takes values in $[0,\tilde{l}\bar\lambda]$ and is measurable with respect to $\mathcal{B}_{[0,T]}\otimes\mathcal{R}$ as a finite sum of measurable maps (claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions).