Poisson process question and more...

**symmetry7** · June 19th 2009, 02:39 AM

Can anyone please help ...

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

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

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

**Laurent** · June 21st 2009, 03:10 AM

Originally Posted by symmetry7

Can anyone please help ...

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

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

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:
$E[Y_t]=\int_0^t (E[B_s]-s)ds=0$ and $E[Y_tY_{t'}]=\int_0^t\int_0^{t'} E[(B_s-s)(B_{s'}-s')]ds' ds$ .
Expand and use $E[B_s B_s']=\min(s,s')$ .

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

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