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 $\lambda$.