Two-jump probability from a bounded hazard

Conditional on the initial history, let $S_1$ be the first jump time. By [F4] its finite-time density $f_1(s)$ is at most $r$. By [F7], conditional on the first jump and its updated full history $h$, the next-event density $f_{2,h}(t)$ is also at most $r$ throughout the remaining interval, because its survivor is at most one. Thus its conditional probability of another jump by $\tau$ is at most $r(\tau -s)$. Integrating the conditional probability with the first-event law yields $\mathrm{Pr}(N_{\tau }-N_0\ge 2)\le {\int }_0^{\tau }rr(\tau -s)ds=r^2{\tau }^2\mathrm{/}2$. This counts the first two events regardless of later events; atoms at infinite waiting time contribute zero, and no independence or stationary intensity is used. The history kernels provide the disintegration used here, so no selection of conditional densities at individual histories is asserted.

The bound in step 1.1 is uniform in the initial history of each check interval, so averaging it gives the same bound for that interval even if earlier measurements change its initial state. By [F9], the probability of at least one interval containing two jumps is at most the sum of the $N$ interval probabilities, namely $N{r}^2{\tau }^2\mathrm{/}2=r^{2}T\tau \mathrm{/}2$. The same argument allows arbitrary state resets between intervals, provided the uniform hazard bound remains valid.

For the specified finite quantum model, [F6] constructs the record process and shows that its ensemble is the channel of [F1], with the finite coarse instruments and normalized conditional states of [F2]. Its conditional hazard is ${\lambda }_t=\mathrm{Tr}(Q{\sigma }_{t-})$. Since ${\sigma }_{t-}$ is positive of trace one and $0\le Q\le rI$, this is between zero and $r$ for every state, including states reachable after an allowed reset. By [F8], the waiting kernel and marked update are jointly measurable functions of the current density matrix, so they can be applied after every event using its updated full history. By [F6], these are the actual conditional laws of the simple non-explosive record process, and its compensator satisfies [F3]. Thus [F7] supplies the history conventions, [F8] supplies the explicit waiting kernels, and [F6] identifies their conditional-law application to this process. The hypotheses used in steps 1.1 and 2.1 therefore hold for this instrument, proving both stated bounds.