Is there an (easy) way to construct, on the same filtered probability space,a Brownian motion $W$ and a Poisson process $N$, such that $W$ and $N$ are not independent ? 


I first asked this question at math.stackexchange.com and was suggested to post it here : <https://math.stackexchange.com/questions/25519/correlated-brownian-motion-and-poisson-process>.


Note that I want $W$ and $N$ to be BM and Poisson for the same filtration $({\cal F}\_t)$ (see Shai Covo's answer on math.stackexchange for a construction without this requirement).



Correlated Brownian motion and Poisson process
You can not construct a Poisson process $N$, and a Brownian motion $W$ on the same filtration such that they are dependent. They are always independent on the given filtration. 


Let $(W\_t)\_{t\geq 0}$ be a standard Brownian motion and $(N\_t)\_{t\geq 0}$ be a Poisson process with intensity $\lambda$, both defined on the same probability space $(\Omega,\mathcal{F},P )$ and with respect to same filtration $F=(\mathcal{F}\_t,t\geq 0)$. Then define the process $L$ for $u\_1,u\_2\in \mathbb{R}$ by


$L\_t:=\exp(u\_1W\_t+u\_2N\_t-\frac{1}{2}u\_1^2t-\lambda(e^{u\_2}-1)t)$. 


Now you can show by using Ito`s lemma for semi-martingales (or jump-diffusions) that $L$ is a martingale and then taking expectation of both sides and using the martingale property, you have 


$E[\exp(u\_1W\_t+u\_2N\_t)]=e^{\frac{1}{2}u\_1^2t}e^{\lambda(e^{u\_2}-1)t}$.


Hence you can conclude that $N\_t$ and $W\_t$ are independent random variables for each $t$, since the last expression involves the product of moment generating function of a normal distribution and a Poisson distribution. 


Edit : @Did, more precise statement would be,


Let $(W\_t)\_{t\geq 0}$ be a standard Brownian motion and $(N\_t)\_{t\geq 0}$ be a Poisson process with intensity $\lambda$, both defined on the same probability space $(\Omega,\mathcal{F},P )$ and with respect to the same filtration $F=(\mathcal{F}\_t,t\geq 0)$. Then the processes $W$ and $N$ are independent.


In the above, only first step is shown, that is, $W\_t$ and $N\_t$ are independent of each for fixed $t$.The next step is to show that for any finite set of times $(t\_1,t\_2,...,t\_n)$, $(W\_{t\_1},...,W\_{t\_n})$ is independent of $(N\_{t\_1},...,N\_{t\_n})$. Hence the assertion follows from this.


