Proof of The Compensated Poisson Process is a Square-Integrable Martingale

Throughout, $\mathbf{1}_{A}$ is $1$ on $A$ and $0$ off $A$; expectations are handled with Linearity and Monotonicity of the Lebesgue Integral; closure and measurability facts (sums, differences with constants, products) are the preliminaries of Square-Integrable Random Variables and the Mean-Square Inner Product, which apply verbatim with any sub-$\sigma$-algebra in place of $\mathcal{F}$; and $\mathcal{F}^{N}_t=\sigma(N_u:0\le u\le t)$ is the natural filtration, to which $N$ is adapted by that definition.

Step 1 (conditions (i) and (ii) of the martingale definition). By condition 3 of Stochastic Process, Independent Increments, and Inhomogeneous Poisson Process with $s=0$ and $N_0=0$, for $t>0$ the variable $N_t=N_t-N_0$ has the Poisson distribution with parameter $\Lambda(t)-\Lambda(0)=\Lambda(t)$; by Moments of the Poisson Distribution it is square-integrable with $\mathbb{E}[N_t]=\Lambda(t)$. For $t=0$, $N_0=0$ is square-integrable. Constants are square-integrable on a probability space, so each $M_t=N_t-\Lambda(t)$ is square-integrable; and $M_t$ is $\mathcal{F}^{N}_t$-measurable, being the difference of the $\mathcal{F}^{N}_t$-measurable variable $N_t$ and a constant. Hence $M$ is a stochastic process, adapted to $(\mathcal{F}^{N}_t)_{t\ge0}$, with square-integrable values: conditions (i) and (ii) of Square-Integrable Martingale, Submartingale, and Supermartingale hold.

Step 2 (the increment is independent of the past). Fix $0\le s<t$ and set $D=N_t-N_s$, a random variable (as recorded in Stochastic Process, Independent Increments, and Inhomogeneous Poisson Process) with the Poisson distribution with parameter $\Lambda(t)-\Lambda(s)$ by condition 3 of that definition; by Moments of the Poisson Distribution, $D$ is integrable with $\mathbb{E}[D]=\Lambda(t)-\Lambda(s)$. We claim that the $\sigma$-algebras $\sigma(D)$ and $\mathcal{F}^{N}_s$ are independent.

Let $\mathcal{P}_s$ consist of $\Omega$ together with all sets of the form $Q=\bigcap_{i=1}^{p}\{N_{u_i}\in B_i\}$ with $p\in\mathbb{N}$, real $0<u_1<\dots<u_p\le s$, and Borel sets $B_i$. This family is a $\pi$-system: intersections of two such sets merge into one after taking the union of the time points and intersecting the Borel sets attached to shared time points (inserting $\mathbb{R}$ at time points missing from one of the two). Every generator $\{N_u\in B\}$ of $\mathcal{F}^{N}_s$ with $0\le u\le s$ lies in $\mathcal{P}_s$ or differs from a member by the constant variable $N_0=0$, whose level sets are $\emptyset$ or $\Omega$; and every member of $\mathcal{P}_s$ lies in $\mathcal{F}^{N}_s$. By minimality of the generated $\sigma$-algebra, $\sigma(\mathcal{P}_s)=\mathcal{F}^{N}_s$.

Fix $Q=\bigcap_{i=1}^{p}\{N_{u_i}\in B_i\}\in\mathcal{P}_s$ and a Borel set $B$; we show the product formula $P(\{D\in B\}\cap Q)=P(D\in B)\,P(Q)$. Consider the grid $0<u_1<\dots<u_p\ (\le s)<t$, refined by inserting $s$ if $u_p<s$, and let $I_1,\dots,I_{q}$ be the increments of $N$ over the consecutive intervals of the grid starting from time $0$, so that $I_q=N_t-N_s=D$ and $q\ge2$. By condition 2 of Stochastic Process, Independent Increments, and Inhomogeneous Poisson Process, the family $I_1,\dots,I_q$ is independent. Index it by $\{1,\dots,q\}\subseteq\mathbb{N}$ and apply Grouping Lemma for Independent Random Variables with the two disjoint blocks $\{1,\dots,q-1\}$ and $\{q\}$: the $\sigma$-algebras $\sigma(I_1,\dots,I_{q-1})$ and $\sigma(D)$ are independent. For each $i$, pointwise $N_{u_i}=I_1+\dots+I_{k(i)}$ where $k(i)$ is the position of $u_i$ in the grid; sums of $\sigma(I_1,\dots,I_{q-1})$-measurable variables are measurable with respect to that $\sigma$-algebra (preliminaries of Square-Integrable Random Variables and the Mean-Square Inner Product applied with this $\sigma$-algebra), so each $\{N_{u_i}\in B_i\}$, and hence $Q$, lies in $\sigma(I_1,\dots,I_{q-1})$. The independence of the two $\sigma$-algebras (product formula of Sigma-Algebra Generated by Random Variables and Independence of Sigma-Algebras with two factors) now gives the claimed product formula; for $Q=\Omega$ it is trivial.

Next, fix a Borel set $B$ and let

Exactly as in Step 3 of the proof of Grouping Lemma for Independent Random Variables (with the single fixed event $\{D\in B\}$ in the role of the fixed intersection there), $\mathcal{L}_B$ is a $\lambda$-system: it contains $\Omega$; it is closed under relative complements $A'\setminus A$ for $A\subseteq A'$ by finite additivity; and it is closed under nondecreasing countable unions by countable additivity and the algebra of limits. It contains $\mathcal{P}_s$ by the previous paragraph, so Dynkin's Pi-Lambda Theorem gives $\mathcal{F}^{N}_s=\sigma(\mathcal{P}_s)\subseteq\mathcal{L}_B$. Since $B$ was arbitrary, and the one-factor cases of the product formula are trivial, the pair $\sigma(D)$, $\mathcal{F}^{N}_s$ is independent in the sense of Sigma-Algebra Generated by Random Variables and Independence of Sigma-Algebras.

Step 3 (the averaged martingale identity). Fix $0\le s<t$ and $A\in\mathcal{F}^{N}_s$. The pair $D$, $\mathbf{1}_{A}$ is independent: for Borel sets $B,B'$, the event $\{D\in B\}$ lies in $\sigma(D)$ and $\{\mathbf{1}_{A}\in B'\}$ is one of $\emptyset,A,\Omega\setminus A,\Omega$, each in $\mathcal{F}^{N}_s$; the required product formulas are instances of the independence established in Step 2 (with $\Omega$ inserted for omitted factors). Both $D$ and $\mathbf{1}_{A}$ are integrable, so Expectation of a Product of Independent Random Variables gives

Pointwise $M_t\mathbf{1}_{A}-M_s\mathbf{1}_{A}=D\mathbf{1}_{A}-\bigl(\Lambda(t)-\Lambda(s)\bigr)\mathbf{1}_{A}$, and all four products are integrable, so linearity and $\mathbb{E}[\mathbf{1}_{A}]=P(A)$ give

For $s=t$ the identity is trivial. Hence the martingale property in averaged form of Square-Integrable Martingale, Submartingale, and Supermartingale holds, which by that definition is equivalent to condition (iii). Together with Step 1, $M$ is a square-integrable martingale with respect to $(\mathcal{F}^{N}_t)_{t\ge0}$. $\blacksquare$