Predictable-Window Moment Identities for the Homogeneous Poisson Process

Let $(\Omega,\mathcal{F},P)$ be a probability space and let $Y=(Y_u)_{u\ge0}$ be a homogeneous Poisson process with rate $1$ on it, all of whose paths are counting paths. For every natural number $p\ge1$ let $\mu_p:[0,\infty)\to[0,\infty]$ be given by $\mu_p(x)=\int_{\mathbb{R}}|t|^{p}\,dP_x(t)$, the Lebesgue integral of the nonnegative measurable map $t\mapsto|t|^{p}$ with respect to the Poisson distribution $P_x$ with parameter $x$. Write $\exp$ for the real exponential function and $\mathbb{E}$ for the expectation, expectations of $[0,\infty]$-valued random variables being their integrals with respect to $P$.

Fix real numbers $\theta\ge0$, $\bar{V}\ge\theta$, and $\bar{\Lambda}\ge0$, and a $\sigma$-algebra $\mathcal{G}\subseteq\mathcal{F}$ that is independent of the $\sigma$-algebra $\sigma(Y_{\theta+s}-Y_{\theta}:\ s\ge0)$ generated by the increments of $Y$ beyond $\theta$. Let $v$ and $\lambda$ be $\mathcal{G}$-measurable random variables with $\theta\le v(\omega)\le\bar{V}$ and $0\le\lambda(\omega)\le\bar{\Lambda}$ for every $\omega$.

1. (Evaluation at random levels) For every random variable $w\ge0$ on $(\Omega,\mathcal{F},P)$ the map $\omega\mapsto Y_{w(\omega)}(\omega)$ is a random variable, and for random variables $0\le w\le w'$ the increment $Y_{w'}-Y_{w}$ is a random variable with values in the nonnegative integers. Moreover, if $\mathcal{G}_0\subseteq\mathcal{F}$ is a $\sigma$-algebra such that $w$ is $\mathcal{G}_0$-measurable and $Y_u$ is $\mathcal{G}_0$-measurable for every real $u\ge0$, then $Y_{w}$ is $\mathcal{G}_0$-measurable.

2. (Window moments) For every natural number $p\ge1$ and every $\mathcal{G}$-measurable $Z:\Omega\to[0,\infty]$, $\mathbb{E}\bigl[Z\,\bigl(Y_{v+\lambda}-Y_{v}\bigr)^{p}\bigr]=\mathbb{E}\bigl[Z\,\mu_p(\lambda)\bigr]\qquad\text{in }[0,\infty].$

3. (Window events) For every $\mathcal{G}$-measurable $Z:\Omega\to[0,\infty]$, $\mathbb{E}\bigl[Z\,\mathbf{1}\{Y_{v+\lambda}-Y_{v}\ge1\}\bigr]=\mathbb{E}\bigl[Z\,\bigl(1-\exp(-\lambda)\bigr)\bigr]\le\mathbb{E}\bigl[Z\,\lambda\bigr]\qquad\text{in }[0,\infty].$

4. (Moment functions) For every natural number $p\ge1$ and every $x\ge0$, $\mu_p(x)=\exp(-x)\sum_{k\ge1}\frac{k^{p}\,x^{k}}{k!}<\infty,$ the sum over the natural numbers $k\ge1$ being well-defined by nonnegativity as the least upper bound of its finite partial sums, with the factorial $k!$; moreover $\mu_1(x)=x$ for every $x\ge0$; each $\mu_p$ is nondecreasing and continuous on $[0,\infty)$; and $\mu_p(x)\le c_p\,x$ for every $x\in[0,1]$, where $c_p=\sum_{k\ge1}k^{p}/k!$ is finite.