Jump-record instruments and conditional intensity

Integrating conjugation by $K_U$ over a measurable cell preserves positivity at every ancilla dimension, since the integral of each nonnegative quadratic form is nonnegative. The integrals exist by the positive trace bound and decomposition into four positive parts. Their sum over a finite partition is the GKSL channel by [F3], hence trace-preserving, so [F4] applies. Conditional on a recorded prefix the remaining law starts from its normalized output: the kernel-product factors in [F2] cancel exactly as in [F1]. This also identifies the conditional state before the next event as the normalized no-jump state. This is the particular instrument chosen by the given $L_a$; the channel alone does not select a unique monitoring instrument.

At a time with no intervening event the survivor from the current state is $s_{\sigma }(u)$ and its derivative is minus the summed marked density. Thus the no-event conditional hazard is $\mathrm{Tr}(Q\widehat{\sigma }(u))$ by [F1]. On the finite-record space of [F2], the conditional state has continuous no-jump evolution between finitely many events and the prescribed right-hand reset at each event; the zero-density reset convention in [F1] defines it on every record. The Borel state maps in [F1], composed along the finite observed prefix, show that the state and its left-limit state are measurable functions of that prefix. Thus $t\mapsto {\sigma }_{t-}$, with ${\sigma }_{0-}=\rho$, is adapted and left-continuous on $(0,U]$. In particular ${\lambda }_t=\mathrm{Tr}(Q{\sigma }_{t-})$, with ${\lambda }_0=\mathrm{Tr}(Q\rho )$, is a real adapted process with left-continuous paths on $(0,U]$ and values in $[0,r]$.

To prove the needed predictability implication locally, let $H$ be any real adapted process with left-continuous paths on $(0,U]$, with $H_0$ measurable for ${\mathcal{F}}_0$. Put $q_k=kU\mathrm{/}{2}^n$ and define $H_t^{(n)}=H_0{\mathbf{1}}_{\{0\}}(t)+{\sum }_{k=0}^{2^n-1}{H}_{q_k}{\mathbf{1}}_{(q_k,q_{k+1}]}(t)$. For each Borel set $C\subseteq \mathbb{R}$, its inverse image is $\{0\}\times \{H_0\in C\}$ together with the finite union of $(q_k,q_{k+1}]\times \{H_{q_k}\in C\}$; all are predictable generators by adaptedness and [F9]. Thus every $H^{(n)}$ is predictable. At $t=0$ these approximants equal $H_0$; for each $t>0$ the selected left endpoints are strictly below $t$ and tend to $t$, so left-continuity gives $H_t^{(n)}\to H_t$ on every record. Pointwise-limit closure in [F9] proves that $H$ is predictable. Apply this to the process $\lambda$ from step 2.1. The constant process $Y_t=1$ is also predictable by the same construction.

By [F9], predictability of $\lambda$ makes its restriction to $[0,t]\times \mathrm{\Omega }$ measurable for $\mathcal{B}([0,t])\otimes {\mathcal{F}}_t$ for each fixed $t$. Apply the section-integral measurability in Tonelli to this nonnegative function on the product of the finite Lebesgue time space and $(\mathrm{\Omega },{\mathcal{F}}_t,P)$; both measures are finite. It follows that $A_t={\int }_0^t{\lambda }_{s}ds$ is ${\mathcal{F}}_t$-measurable. The pathwise bound $0\le \lambda \le r$ gives $\mathrm{\mid }{A}_t-A_s\mathrm{\mid }\le r\mathrm{\mid }t-s\mathrm{\mid }$, so $A$ is adapted and continuous on every record, with $A_0=0$, $A_U\le rU$ and $\mathbb{E}[A_U]\le rU$. Apply the local approximation of step 3.1 to $H=A$, including its deterministic time-zero value, to conclude that $A$ is predictable.

Here is an explicit verification of the martingale requirement, rather than an inference from the hazard formula alone. Starting from any state at time $s$, let $Z_n(v)$ be the integral over exactly $n$ events in $(s,v)$ of their unnormalized record state propagated without further events up to $v$. The expected rate density of the next event after these $n$ events is $\mathrm{Tr}(Q{Z}_n(v))$: integrate and sum the last marked jump and then all continuations after it, whose conditional mass is one by [F2]. Consequently $\mathbb{E}[N_u-N_s]={\int }_s^u{\sum }_{n\ge 0}\mathrm{Tr}(Q{Z}_n(v))dv$. On the other hand the weight of each $n$-event state at $v$ is its trace, and multiplying that weight by its normalized hazard cancels the normalization. Hence $\mathbb{E}[{\lambda }_v]={\sum }_{n\ge 0}\mathrm{Tr}(Q{Z}_n(v))$. Tonelli justifies both identities, and the sums are bounded by $r$, since the trace weights sum to one. The same computation applies conditional on every prefix at $s$, by consistency and the explicit kernels. Integrating against an arbitrary bounded measurable function of that prefix, and using Fubini, gives $\mathbb{E}[N_u-N_s\mid {\mathcal{F}}_s]=\mathbb{E}[{\int }_s^u{\lambda }_{v}dv\mid {\mathcal{F}}_s]$ for each fixed $s\le u$. Thus $N-A$ is an all-pairs martingale.

For an integer-valued nonnegative variable $N_U$, $N_U^2={\sum }_{n=1}^{N_U}(2n-1)$. By the tail bound and Tonelli, $\mathbb{E}[N_U^2]\le {\sum }_{n\ge 1}(2n-1)(rU{)}^n\mathrm{/}n!<\mathrm{\infty }$; convergence follows by the ratio test, or by the exponential series. Since $A_U\le rU$, every $N_t-A_t$ is square-integrable. Together with step 4.1 and the martingale identity, this establishes every clause of [F5].