A finite-window Poisson process that terminates at time one

The extension by zero has unit jumps only at the simple event times of $\eta$, and $N_t$ is finite because $N_1$ is Poisson with mean $1$. The effective intensity ${\lambda }^{\ast }(t)={\mathbf{1}}_{(0,1]}(t)$ is deterministic Borel and hence predictable by [F4]. Direct integration gives $A_t={\int }_0^t{\lambda }^{\ast }(u)du=t\wedge 1$, so $A$ is continuous, nondecreasing, and bounded by $1$.

Let $\mathsf{H}$ be the space of finite counting measures on $(0,\mathrm{\infty })$ with the sigma-algebra generated by its Borel-set evaluation maps. Define the history random element by $H_s(B):=\eta (B\cap (0,s\wedge 1])$ for every Borel $B\subseteq (0,\mathrm{\infty })$. It is a finite counting measure, and it is measurable because each evaluation is a measurable count of the supplied point process; its generated sigma-algebra is the event-history filtration in the Given clause. For finite $u>0$, define $h\oplus u:=h+{\delta }_u$. For every Borel $B$, $(h\oplus u)(B)=h(B)+{\mathbf{1}}_B(u)$, so the update is jointly measurable in $h,u$ and appends exactly one event.

For $0\le s\le t$, the increment $N_t-N_s$ is the count on $(s,t]\cap (0,1]$, a Poisson variable of mean $\min (t,1)-\min (s,1)$. Refine a finite family of past Borel sets into the disjoint atoms of the finite Boolean algebra they generate. Every atom lies in $(0,s]$, so [F1] gives independence of its count vector from the future increment. These count cylinders generate the natural-history sigma-algebra, and [F5] extends independence to that sigma-algebra. Therefore $\mathbb{E}[N_t-N_s\mid {\mathcal{H}}_s]=\min (t,1)-\min (s,1)=A_t-A_s$. Thus $N-A$ is a martingale. Since $N_t$ has finite second moment by [F3] and $A_t\le 1$, this martingale is square-integrable and meets [F6]; $\mathbb{E}[A_T]\le 1$.

For $s\ge 0$ and $h\in \mathsf{H}$, supply the next-event kernel, viewed on $[0,\mathrm{\infty }]$ and supported on $(s,\mathrm{\infty }]$, by $K_{s,h}(C):={\int }_{C\cap (s,1]}{e}^{-(u-s)}du+e^{-(1-s{)}_{+}}{\mathbf{1}}_{\{\mathrm{\infty }\in C\}}$ for Borel $C\subseteq [0,\mathrm{\infty }]$, where $x_{+}:=\max (x,0)$. Its finite-time mass is $1-e^{-(1-s{)}_{+}}$, so each $K_{s,h}$ is a probability measure. For fixed $C$, the finite-time term is $e^s{\int }_{C\cap (s,1]}{e}^{-u}du$, a Borel function of $s$ because the tail integral is monotone on $[0,1]$ and zero for $s\ge 1$; the atom mass is also Borel. The finite-time density $f_{s,h}(u)={\mathbf{1}}_{\{s<u\le 1\}}{e}^{-(u-s)}$ is jointly measurable in $(s,h,u)$. By independent future increments, this is the conditional next-event law given $H_s$: its survivor for $u>s$ is $S_{s,h}(u)=e^{-((u\wedge 1)-s{)}_{+}}$, since the event of no point in $(s,u]\cap (0,1]$ has probability $e^{-((u\wedge 1)-s{)}_{+}}$. Thus $f_{s,h}\mathrm{/}{S}_{s,h}=1$ on $(s,1]$ and $0$ after $1$, agreeing almost everywhere with ${\lambda }^{\ast }(u\mid h_{u-})={\mathbf{1}}_{(0,1]}(u)$; this also agrees with [F7]. In particular, $K_{0,h}(\{\mathrm{\infty }\})=e^{-1}$ and $K_{s,h}(\{\mathrm{\infty }\})=1$ for $s\ge 1$.

Taking $u\to \mathrm{\infty }$ in step 3.1 gives $\mathrm{Pr}(T_1=\mathrm{\infty })=e^{-1}$, also the zero-count probability $\mathrm{Pr}(N_1=0)$ by [F3]. No events occur after time $1$, and $N_1$ is finite almost surely, so the process terminates and is non-explosive.