I would like to show that « Poisson random processes » and « Poisson random measures » are the same objects. More precisely, suppose that $N_t$ is a Poisson random process with intensity $\lambda$ on some probability space i.e. :
- $N_t$ is an increasing right-continuous integer-valued random process such that $N_0 = 0$
- $N_t$ has independent and stationary increments
- $N_t$ has a Poisson distribution with parameter $\lambda t$
We can identify $N_t$ with its (random) Lebesgue-Stieltjes measure $N$, defined for any Borel set $B \subset [0,\infty)$ by $$ N(B) = \int 1_{B}(s)\, \mathrm dN_s $$ How to prove that $N$ is a random Poisson measure with intensity $\lambda\,\mathrm ds$ in the sense that :
- Whenever $B_1,\ldots,B_n$ are disjoint Borel sets, the random variables $N(B_1),\ldots,N(B_n)$ are independent.
- Whenever $B$ has finite Lebesgue measure, $N(B)$ has a Poisson distribution with parameter $\lambda \int 1_B(s)\,\mathrm ds$.
My attempt was to prove a well-known formula about Poisson random measures : for all nonnegative Borel-measurable function $u$ $$ \mathbb E \exp \bigg\{- \int u(x)\,\mathrm dN \bigg\} = \exp \bigg\{- \int (1-e^{-u})\,\mathrm d\mu\bigg\} $$ where $\mu$ denotes the intensity measure, here $\mathrm d\mu = \lambda\,\mathrm ds$. I was able to prove this formula for $u$ piecewise continuous with compact support... but I don't know how to go further.
Any hint would be appreciated.
Partial Answer
Let's write $$ \Delta N_t=N_t-N_{t-} $$ for the jump of $N$ at $t$. For each bounded set $B\subset[0,\infty)$ the Poisson process $N$ has only finitely many jumps in $t\in B$. Further, $$ \Delta N_{t_1},...,\Delta N_{t_n} $$ are independent for each finite set $\{t_1,...,t_n\}$ because $N$ has independent increments.
When $B$ is a bounded Borel set then we note that $$ N(B)=\sum_{s\in B}\Delta N_s. $$ is a finite sum of jumps. This should imply that $$ N(B_1),....,N(B_n) $$ are independent for each collection of disjoint bounded Borel sets $B_1,...,B_n$.