The Poisson process

Suppose $P_{t}$ is a Poisson process and we write $X_{t}=P_{t-}$. Is $P_{1}-X_{1-t}$ a Poisson process on $[0,1]$ ? Why or why not?

Let $P$ be a Poisson process with parameter $\lambda$. Show that

Show that if $P^{(1)}$ and $P^{(2)}$ are independent Poisson processes with parameters $\lambda_{1}$ and $\lambda_{2}$, respectively, then $P_{t}^{(1)}+P_{t}^{(2)}$ is a Poisson process with parameter $\lambda_{1}+\lambda_{2}$.

If $X$ is defined by (5.2), show that $X$ is a Poisson process.

Let $X_{t}$ be a stochastic process and let $\left\{ \mathcal{F}_{t}^{00}\right\}$ be the filtration generated by $X$. Suppose $X$ is a Poisson process with respect to the filtration $\left\{ \mathcal{F}_{t}^{00}\right\}$. Show that $X$ is a Poisson process with respect to the minimal augmented filtration generated by $X$.

Hint: Imitate the proof of Proposition 2.5 .

Suppose $P_{t}$ is a Poisson process and $f$ and $g$ are non-negative bounded deterministic functions with compact support. Find necessary and sufficient conditions on $f$ and $g$ so that $\int_{0}^{\infty } f(s) d P_{s}$ and $\int_{0}^{\infty } g(s) d P_{s}$ are independent.

Hint: First show that the characteristic function of $F=\int_{0}^{\infty } f(s) d P_{s}$ is

We will talk about weak convergence in general metric spaces in Chapters 30-35. This exercise is concerned with the weak convergence of real-valued random variables as defined in Section A. 12 .

Suppose for each $n, P_{n}$ is a Poisson random variable with parameter $\lambda_{n}$ and $\lambda_{n} \rightarrow \infty$ as $n \rightarrow \infty$. Prove that

converges weakly to a normal random variable with mean zero and variance one.

Hint: Imitate the proof of Theorem A.51.

Assuming there is no difficulty with the convergence, we have

Exercise 6.2 asks you to verify that the process $W$ defined by (6.8) is a mean zero Gaussian process on $[0,1]$ with the same covariances as a Brownian motion.