# Thread: Poisson process question and more...

1. ## Poisson process question and more...

Let $\displaystyle {X}(t)=\int_{0}^{t}\tau{d}{N}(\tau)$, where $\displaystyle {N}(\tau)$ is a Poisson process with rate $\displaystyle \lambda$.
Find $\displaystyle {E}{X}(t)$, and $\displaystyle cov\{{X}(t),{X}(\tau)\}$.

Also, let $\displaystyle {B}_{s}$ be Brownian motion and $\displaystyle {Y}(t)=\int_{0}^{t}({B}_{s}-{s})ds$,
Find $\displaystyle {E}{Y}(t)$, and $\displaystyle cov\{{Y}(t),{Y}(\tau)\}$.

I'm having real trouble with these, so if anyone could give concise details of steps involved with explanations, that would be great.

2. Originally Posted by symmetry7

Let $\displaystyle {X}(t)=\int_{0}^{t}\tau{d}{N}(\tau)$, where $\displaystyle {N}(\tau)$ is a Poisson process with rate $\displaystyle \lambda$.
Find $\displaystyle {E}{X}(t)$, and $\displaystyle cov\{{X}(t),{X}(\tau)\}$.

Also, let $\displaystyle {B}_{s}$ be Brownian motion and $\displaystyle {Y}(t)=\int_{0}^{t}({B}_{s}-{s})ds$,
Find $\displaystyle {E}{Y}(t)$, and $\displaystyle cov\{{Y}(t),{Y}(\tau)\}$.

I'm having real trouble with these, so if anyone could give concise details of steps involved with explanations, that would be great.
The second one is a bit simpler, you only have to apply Fubini:
$\displaystyle E[Y_t]=\int_0^t (E[B_s]-s)ds=0$ and $\displaystyle E[Y_tY_{t'}]=\int_0^t\int_0^{t'} E[(B_s-s)(B_{s'}-s')]ds' ds$.
Expand and use $\displaystyle E[B_s B_s']=\min(s,s')$.

For the first one, one way is to reduce to the same method as for the first one: by integration by part, $\displaystyle X(t)=tN(t)-\int_0^t N(\tau)d\tau = \int_0^t (N(t)-N(\tau))d\tau$. Then do like above (but the expectation is nonzero).