Proof of The Compensated Poisson Process is an Ito Integrator with Its Intensity

Throughout, $\lambda$ denotes Lebesgue measure and $\lambda_{\mathrm{int}}$ the intensity function of the statement. We first record: for a real constant $c$, the translation $x\mapsto x-c$ is continuous, hence measurable for the Borel $\sigma$-algebra (preimages of open sets are open, and open sets generate), so for every Borel set $B$ the translate $B+c=\{x+c:x\in B\}$, being the preimage of $B$ under $x\mapsto x-c$, is Borel.

Step 1 (Clauses (i) and (ii)). By The Compensated Poisson Process is a Square-Integrable Martingale, $M$ is a square-integrable martingale with respect to $(\mathcal{F}^{N}_t)_{t\ge0}$. By clause 1 of Stochastic Process, Independent Increments, and Inhomogeneous Poisson Process, $N_0=0$ and $\Lambda(0)=0$, so $M_0=0$ identically, in particular almost surely.

Step 2 (Clause (iii)). $N$ has independent increments and $N_0=0$ is constant, so Increments Are Independent of the Natural Filtration Past shows that $\sigma(N_t-N_s)$ and $\mathcal{F}^{N}_s$ are independent $\sigma$-algebras for all $0\le s<t$. With $c=\Lambda(t)-\Lambda(s)$ we have $M_t-M_s=(N_t-N_s)-c$, hence for every Borel $B$,

so every member of $\sigma(M_t-M_s)$ lies in $\sigma(N_t-N_s)$ (single-random-variable description in Sigma-Algebra Generated by Random Variables and Independence of Sigma-Algebras). A sub-$\sigma$-algebra of a $\sigma$-algebra independent of $\mathcal{F}^{N}_s$ is itself independent of $\mathcal{F}^{N}_s$, directly from the definition. This is clause (iii) of Ito Integrator of Intensity Type.

Step 3 (Clause (iv)). The intensity $\lambda_{\mathrm{int}}$ is nonnegative and continuous on every interval $[0,T]$ by Stochastic Process, Independent Increments, and Inhomogeneous Poisson Process; its extension by $0$ to $\mathbb{R}$ is Borel measurable, since for $a\ge0$ the superlevel set $\{\lambda_{\mathrm{int}}>a\}$ is relatively open in $[0,\infty)$ (continuity on every $[0,T]$), hence the intersection of an open set with the Borel set $[0,\infty)$, while for $a<0$ the superlevel set is $\mathbb{R}$.

Fix $0\le s<t$. By agreement of the Riemann and Lebesgue integrals for the continuous function $\lambda_{\mathrm{int}}$ on $[s,t]$, and since the indicators of $[s,t]$ and $(s,t]$ differ only at the single point $s$ with $\lambda(\{s\})=0$ (a degenerate interval has length $0$, and the integral of a nonnegative function over a null set vanishes, cf. the null-set remark in Square-Integrable Random Variables and the Mean-Square Inner Product applied via monotonicity of the integral, Linearity and Monotonicity of the Lebesgue Integral),

the Riemann integral, using the additivity of the mean function recorded in Stochastic Process, Independent Increments, and Inhomogeneous Poisson Process. In particular $\int_{\mathbb{R}}\mathbf{1}_{(0,T]}\lambda_{\mathrm{int}}\,d\lambda=\Lambda(T)<\infty$ for every $T>0$.

By clause 3 of Stochastic Process, Independent Increments, and Inhomogeneous Poisson Process, $N_t-N_s$ has the Poisson distribution with parameter $\mu=\Lambda(t)-\Lambda(s)$, so by Moments of the Poisson Distribution it is square-integrable with $\mathbb{E}[N_t-N_s]=\mu$ and variance $\operatorname{Var}(N_t-N_s)=\mu$. Hence

This is clause (iv) of Ito Integrator of Intensity Type, so $(M,\lambda_{\mathrm{int}})$ is an It^{o} integrator of intensity type with respect to $(\mathcal{F}^N_t)_{t\ge0}$.

Step 4 (Equality of natural filtrations). Fix $t\ge0$. For $u\le t$ and Borel $B$, with $c_u=\Lambda(u)$:

using that translates of Borel sets are Borel. Thus each generating family of $\mathcal{F}^{M}_t$ (respectively $\mathcal{F}^{N}_t$) is contained in the other $\sigma$-algebra, and by minimality of generated $\sigma$-algebras, $\mathcal{F}^{M}_t=\mathcal{F}^{N}_t$. $\square$