A question about Poisson process (waiting online)

[ssyldy]

Hey guys, I encounter a question (maybe a silly one )that puzzles me. N_t is a Poisson process and λ is the jump intensity.Since the quadratic variation of Poisson process is [N,N]_t=N_t, and N_t²-[N,N]_t is a martingale, it follows that E[N_t²]=E[[N,N]_t]=λ*t. On the other hand, the direct calculation of E[N_t²] can be found in https://proofwiki.org/wiki/Variance_of_Poisson_Distribution, which indicates E[N_t²]=(t*λ)²+t*λ. The two results are different. I really appreciate it if somebody can help me.

 

[mfb]

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

 

[ssyldy]

N_t ^{is a Poisson process and λ is the jump intensity.Since the quadratic variation of Poisson process is [N,N]_t=N_t, and N_t2-[N,N]_t is a martingale, it follows that E[N_t2]=E[[N,N]_t]=λ*t. On the other hand, the direct calculation of E[N_t2] can be found in https://proofwiki.org/wiki/Variance_of_Poisson_Distribution}

 

[mathman]

Hey guys, I encounter a question (maybe a silly one )that puzzles me. N_t is a Poisson process and λ is the jump intensity.Since the quadratic variation of Poisson process is [N,N]_t=N_t, and N_t²-[N,N]_t is a martingale, it follows that E[N_t²]=E[[N,N]_t]=λ*t. On the other hand, the direct calculation of E[N_t²] can be found in https://proofwiki.org/wiki/Variance_of_Poisson_Distribution, which indicates E[N_t²]=(t*λ)²+t*λ. The two results are different. I really appreciate it if somebody can help me.

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

 

[ssyldy]

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

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

 

[mathman]

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

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

 

[ssyldy]

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

Yeah, I know that. But my question is, why would I get two different results of E[N_t²] using two different methods?

 

[mathman]

Yeah, I know that. But my question is, why would I get two different results of E[N_t²] using two different methods?

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