Likelihood of a Causal Intensity on the Observation Record Space

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 and channel set $V=\{1,\dots,\tilde{l}\}$, and let $\lambda=(\lambda^\upsilon)_{\upsilon\in V}$ be a causal intensity on $\mathbf{R}$ with bound $\bar\lambda$ and total intensity $\lambda^{\mathrm{tot}}$. For every $r\in\mathbf{R}$ the map $s\mapsto\lambda^{\mathrm{tot}}_s(r)$ on $[0,T]$ is measurable with respect to the trace Borel $\sigma$-algebra (it is a section of a map measurable with respect to the product $\sigma$-algebra, and the sets whose sections at $r$ are measurable form a $\sigma$-algebra containing the measurable rectangles) and takes values in $[0,\tilde{l}\bar\lambda]$, so its Lebesgue integral over $[0,T]$ is a finite real number.

The likelihood of $\lambda$ is the function $\ell_\lambda:\mathbf{R}\to[0,\infty)$ given, for $r=(k,t,v)$ with $t=(t_1,\dots,t_k)$ and $v=(v_1,\dots,v_k)$, by $\ell_\lambda(r)=\Bigl(\prod_{i=1}^{k}\lambda^{v_i}_{t_i}(r)\Bigr)\exp\Bigl(-\int_{[0,T]}\lambda^{\mathrm{tot}}_s(r)\,ds\Bigr),$ with the finite product over $i$ equal to $1$ when $k=0$, and $\exp$ the real exponential function.