Poisson point processes

Suppose $\left\{ \mathcal{F}_{t}\right\}$ is a filtration satisfying the usual conditions and $P_{t}^{1}$ and $P_{t}^{2}$ are Poisson processes with respect to $\left\{ \mathcal{F}_{t}\right\}$ with parameters $\lambda_{1}, \lambda_{2}$, respectively. Suppose $P_{t}^{1}+P_{t}^{2}$ is a Poisson process with parameter $\lambda_{1}+\lambda_{2}$. Prove that $P^{1}$ and $P^{2}$ are independent processes.

Suppose $\left\{ \mathcal{F}_{t}\right\}$ is a filtration satisfying the usual conditions, $P_{t}$ is a Poisson process with respect to $\left\{ \mathcal{F}_{t}\right\}$, and $W_{t}$ is a Brownian motion with respect to $\left\{ \mathcal{F}_{t}\right\}$. Show that if $W_{t}+P_{t}$ has stationary and independent increments, then $P$ and $W$ are independent processes.

If $H$ is as in (18.1) and $N$ is defined by (18.2), show that $N$ is a martingale, $N^{c}=0$, and $[N]_{t}$ is given by (18.3).

Suppose $\left\{ A_{s}, 0<s<\infty \right\}$ is a collection of subsets of $\mathcal{S}$ such that $\lambda \left(A_{s}\right) \rightarrow \infty$ as $s \rightarrow \infty$. Show that $N_{t}\left(A_{s}\right) / \lambda \left(A_{s}\right)$ converges to $t$ uniformly over finite intervals, where the convergence is in probability.

Suppose $\left\{ A_{s}, 0<s<\infty \right\}$ is a collection of subsets of $\mathcal{S}$ such that $A_{r} \subset A_{s}$ if $r \leq s$ and $\lambda \left(A_{s}\right) \rightarrow \infty$ as $s \rightarrow \infty$. Show that for each $t$,

tends to zero almost surely as $s \rightarrow \infty$.

Let $\mathcal{S}$ be a metric space and $\lambda$ a $\sigma$-finite measure on $\mathcal{S}$. Construct a Poisson point process which has $\lambda$ as the corresponding measure.

Let $P_{t}^{j}, j=1,2, \ldots$ be independent Poisson processes with parameter $\beta_{j}$. Let $X_{t}=\sum_{j=1}^{\infty } a_{j} P_{t}^{j}$, where $a_{j}$ is a sequence such that $X_{t}$ is finite, a.s. For $A \subset \mathbb {R} \backslash \left\{ 0\right\}$, define $N_{t}(A)$ to be the number of times before time $t$ that $X$ has a jump whose size is in $A$ :

Prove that $N_{t}$ is a Poisson point process and determine $\lambda$.

Let $\delta >0$ and $A_{\delta }=\left\{ f \in \mathcal{E}: \sup_{t}\left|f(t)\right|>\delta \right\}$. Let $S_{1}=\inf \left\{ t: K_{t}\left(A_{\delta }\right)=1\right\}$ and $S_{2}=\inf \left\{ t>S_{1}: K_{t}\left(A_{\delta }\right)=2\right\}$. Thus $S_{1}$ and $S_{2}$ are the times the first and second excursions in $A_{\delta }$ have been completed. Let $Y_{1}(t)$ be the excursion completed at time $S_{1}$ and define $Y_{2}(t)$ similarly. To be more precise, if $R_{1}=\sup \left\{ t<S_{1}: W_{t}=0\right\}$, then $Y_{1}(s)=W_{R_{1}+s}$ if $s \leq S_{1}-R_{1}$ and $Y_{1}(s)$ is equal to 0 for all $s \geq S_{1}-R_{1}$.

Prove that $Y_{1}$ and $Y_{2}$ are independent.

Hint: Use the strong Markov property at time $S_{1}$.

Let $N_{t}(A)$ be a Poisson point process with respect to the measure space $(\mathcal{S}, m)$ and let $A_{s}, s>0$, be an increasing sequence of subsets of $\mathcal{S}$ with $m\left(A_{s}\right) \rightarrow \infty$ as $s \rightarrow \infty$. Does

converge weakly with respect to $D[0,1]$ as $s \rightarrow \infty$ ? What is the limit?

This can be applied to get central limit theorems for the number of downcrossings of a Brownian motion, for example.

Let $X$ be a Lévy process with Lévy measure $m$. Prove that if $A$ and $B$ are disjoint closed sets, then

for each $x$, where $B-y=\left\{ z-y: z \in B\right\}$. This is the Lévy system formula in the case of Lévy processes. There is an analogous formula for Hunt processes.

Here is an application of the existence theorem in which $T$ is not a subset of the line. Let $(N, \mathscr {N}, \nu )$ be a measure space, and take $T$ to consist of the $\mathscr {N}$-sets of finite $\nu$-measure. The problem is to construct a generalized Poisson process, a stochastic process $\left[X_{A}: A \in T\right]$ such that (i) $X_{A}$ has the Poisson distribution with mean $\nu (A)$ and (ii) $X_{A_{1}}, \ldots , X_{A_{n}}$ are independent if $A_{1}, \ldots , A_{n}$ are disjoint. Hint: To define the finite-dimensional distributions, generalize this construction: For $A, B$ in $T$, consider independent random variables $Y_{1}, Y_{2}, Y_{3}$ having Poisson distributions with means $\nu \left(A \cap B^{c}\right), \nu (A \cap B), \nu \left(A^{c} \cap B\right)$, take $\mu_{A, B}$ to be the distribution of $\left(Y_{1}+Y_{2}, Y_{2}+Y_{3}\right)$.

Example 6. Let $\lambda$ be a finite measure ($\lambda \not\equiv 0$) defined on some measurable space $(\mathrm{S}, \mathscr {B})$. Let $Y, X_{1}, X_{2}, \ldots$ be independent random elements on a probability space $(\Omega , \mathscr {F}, \mathbb {P})$ such that $Y$ is a real Poisson random variable with parameter $\lambda (\mathrm{S})$, and $X_{1}, X_{2}, \ldots$ are independent identically distributed $\mathscr {F} \mid \mathscr {B}$-measurable variables for which $\mathbb {P}_{X_{1}}\left(B\right) = \lambda (B) / \lambda (\mathrm{S})$ for $B \in \mathscr {B}$. Define on $\mathscr {B} \times \Omega$ the function $Z(B, \omega )$ by setting

($Z(B, \omega ) = 0$ when $Y(\omega ) = 0$). Formula (14) defines the so-called Poisson random measure with intensity measure (or leading measure) $\lambda$.

Let $\lambda$ be a $\sigma$-finite measure on some measurable space ( $\mathrm{S}, \mathscr {B}$ ) $\left(\mathrm{S} = \bigcup_{m = 1}^{\infty } \mathrm{S}_{m}\right.$, where $\mathrm{S}_{m} \in \mathscr {B}, \mathrm{S}_{m} \cap \mathrm{S}_{k} = \varnothing$ for $m \neq k$ and $0 < \lambda \left(\mathrm{S}_{m}\right) < \infty$, $m, k \in \mathbb {N}$ ). In this case the construction of the Poisson random measure (14) is easily modified as follows.

On some probability space $(\Omega , \mathscr {F}, \mathbb {P})$ construct a countable collection of independent variables

such that $Y^{(m)}$ has the Poisson distribution with parameter $\lambda_{m} = \lambda \left(\mathrm{S}_{m}\right)$ and $\operatorname {Law}\left(X_{j}^{(m)}\right) = \lambda (\cdot ) / \lambda \left(\mathrm{S}_{m}\right)$ on $\mathscr {B}_{m} = \mathscr {B} \cap \mathrm{S}_{m} ; m, j \in \mathbb {N}$. Let $Z_{m}(B, \omega )$ for $B \in \mathscr {B}_{m}$ be defined according to (14) with $Y$ replaced by $Y^{(m)}$ and $X_{j}$ by $X_{j}^{(m)}$. Set

Note that $Z(B, \omega )$ may also take infinite values.

Prove that if $\lambda (B) = \infty$ for some $B \in \mathscr {B}$, then $Z(B) = \infty$ a.s., and if $\lambda (B) < \infty$, then $Z(B) < \infty$ a.s.

Example 6. Let $\lambda$ be a finite measure ($\lambda \not\equiv 0$) defined on some measurable space $(\mathrm{S}, \mathscr {B})$. Let $Y, X_{1}, X_{2}, \ldots$ be independent random elements on a probability space $(\Omega , \mathscr {F}, \mathbb {P})$ such that $Y$ is a real Poisson random variable with parameter $\lambda (\mathrm{S})$, and $X_{1}, X_{2}, \ldots$ are independent identically distributed $\mathscr {F} \mid \mathscr {B}$-measurable variables for which $\mathbb {P}_{X_{1}}\left(B\right) = \lambda (B) / \lambda (\mathrm{S})$ for $B \in \mathscr {B}$. Define on $\mathscr {B} \times \Omega$ the function $Z(B, \omega )$ by setting

($Z(B, \omega ) = 0$ when $Y(\omega ) = 0$). Formula (14) defines the so-called Poisson random measure with intensity measure (or leading measure) $\lambda$.

Let $\lambda$ be a $\sigma$-finite measure on some measurable space ( $\mathrm{S}, \mathscr {B}$ ) $\left(\mathrm{S} = \bigcup_{m = 1}^{\infty } \mathrm{S}_{m}\right.$, where $\mathrm{S}_{m} \in \mathscr {B}, \mathrm{S}_{m} \cap \mathrm{S}_{k} = \varnothing$ for $m \neq k$ and $0 < \lambda \left(\mathrm{S}_{m}\right) < \infty$, $m, k \in \mathbb {N}$ ). In this case the construction of the Poisson random measure (14) is easily modified as follows.

On some probability space $(\Omega , \mathscr {F}, \mathbb {P})$ construct a countable collection of independent variables

such that $Y^{(m)}$ has the Poisson distribution with parameter $\lambda_{m} = \lambda \left(\mathrm{S}_{m}\right)$ and $\operatorname {Law}\left(X_{j}^{(m)}\right) = \lambda (\cdot ) / \lambda \left(\mathrm{S}_{m}\right)$ on $\mathscr {B}_{m} = \mathscr {B} \cap \mathrm{S}_{m} ; m, j \in \mathbb {N}$. Let $Z_{m}(B, \omega )$ for $B \in \mathscr {B}_{m}$ be defined according to (14) with $Y$ replaced by $Y^{(m)}$ and $X_{j}$ by $X_{j}^{(m)}$. Set

Note that $Z(B, \omega )$ may also take infinite values.

(see [32; p. 370]). Let $B \in \mathscr {B}$ be such that $0 < \lambda (B) < \infty$, where $\lambda$ is a $\sigma$-finite measure on $(\mathrm{S}, \mathscr {B})$. Let $B = \bigcup_{q = 1}^{n} B_{q}$, where $B_{q} \in \mathscr {B}, B_{j} \cap B_{q} = \varnothing$ for $q \neq j(q, j = 1, \ldots , n)$. Prove that for any $k_{1}, \ldots , k_{n} \in \mathbb {Z}_{+}$ and $k = \sum_{q = 1}^{n} k_{q}$ the following equality holds

Define $m(B) = \mathbb {E}\left[\mu (B, \cdot )\right]$ for all $B \in \mathscr {B}\left(\mathbb {R}_{+}\right)$, where $\mu (B, \cdot )$ appears in (85).

Suppose that on some probability space $(\Omega , \mathscr {F}, \mathbb {P})$ we are given a sequence of random variables $\left\{ \tau_{k}\right\}_{k \geqslant 1}$ such that for $\omega \in \Omega_{0}$, where $\mathbb {P}\left(\Omega_{0}\right) = 1$,

Definition 11. The random point measure $\mu : \mathscr {B}\left(\mathbb {R}_{+}\right) \times \Omega \rightarrow \overline{\mathbb {Z}}_{+} = \mathbb {Z}_{+} \cup \left\{ \infty \right\}$ is given by the formula

where the sequence $\left\{ \tau_{k}\right\}_{k \geqslant 1}$ is described above, and $\delta_{x}$ is the Dirac measure.

Will the function $m(\cdot )$ be a measure on $\mathscr {B}\left(\mathbb {R}_{+}\right)$? Can a Poisson process with a locally finite intensity measure $m$ be represented in the form (86)?

Definition 12. The random point process $Y = \left\{ Y(t), t \geqslant 0\right\}$, where $Y(0) = 0$, is introduced for $t > 0$ as follows:

where the random point measure $\mu$ is given by formula (85).

Let $N = \left\{ N(t), t \geqslant 0\right\}$ be a Poisson process with mean measure $m = m(\cdot )$, determined by the sequence $\left\{ \tau_{k}\right\}_{k \geqslant 1}$ that specifies the jump points of $N$. Let $\left\{ \eta_{k}\right\}_{k \geqslant 1}$ be a sequence of independent identically distributed real random variables with $\operatorname {Law}\left(\eta_{k}\right) = \widetilde{\mathbb {P}}$. Take $D \in \mathscr {B}(\mathbb {R})$ such that $\widetilde{\mathbb {P}}(D) > 0$. Prove that the "thinned process" $Y(t, D), t \geqslant 0$, defined by formula (88), is a Poisson process with mean measure $m(\cdot ) \widetilde{\mathbb {P}}(D)$.

Definition 11. The random point measure $\mu : \mathscr {B}\left(\mathbb {R}_{+}\right) \times \Omega \rightarrow \overline{\mathbb {Z}}_{+} = \mathbb {Z}_{+} \cup \left\{ \infty \right\}$ is given by the formula

where the sequence $\left\{ \tau_{k}\right\}_{k \geqslant 1}$ is described above, and $\delta_{x}$ is the Dirac measure.

Definition 12. The random point process $Y = \left\{ Y(t), t \geqslant 0\right\}$, where $Y(0) = 0$, is introduced for $t > 0$ as follows:

where the random point measure $\mu$ is given by formula (85).

Formula (88):

where the process $Y$ is given by formula (86).