A question about Poisson process (waiting online)

1. Jul 19, 2015

ssyldy

Hey guys, I encounter a question (maybe a silly one )that puzzles me. Nt is a Poisson process and λ is the jump intensity.Since the quadratic variation of Poisson process is [N,N]t=Nt, and Nt2-[N,N]t is a martingale, it follows that E[Nt2]=E[[N,N]t]=λ*t. On the other hand, the direct calculation of E[Nt2] can be found in https://proofwiki.org/wiki/Variance_of_Poisson_Distribution, which indicates E[Nt2]=(t*λ)2+t*λ. The two results are different. I really appreciate it if somebody can help me.

Last edited: Jul 19, 2015
2. Jul 19, 2015

Staff: Mentor

It would help to know the context of that calculation and the meaning of the individual symbols.

3. Jul 19, 2015

ssyldy

Nt is a Poisson process and λ is the jump intensity.Since the quadratic variation of Poisson process is [N,N]t=Nt, and Nt2-[N,N]t is a martingale, it follows that E[Nt2]=E[[N,N]t]=λ*t. On the other hand, the direct calculation of E[Nt2] can be found in https://proofwiki.org/wiki/Variance_of_Poisson_Distribution

4. Jul 19, 2015

mathman

It looks like someone has confused the second moment with the variance.

5. Jul 19, 2015

ssyldy

Hi dude, seems like you know the answer. Could you explain?

6. Jul 20, 2015

mathman

General formulas for a random variable X:
second moment: $E(X^2)$
variance: $E(X^2)-(E(X))^2$

7. Jul 20, 2015

ssyldy

Yeah, I know that. But my question is, why would I get two different results of E[Nt2] using two different methods?

8. Jul 21, 2015

mathman

My guess - a mistake in the derivation. $E[N_t^2]=\lambda t$ looks wrong (unless the mean=0). It is the variance.