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Let $\{N_t|0<t\leqslant T \}$ and $\{M_t|0<t\leqslant T \}$ be two Poisson processes with intensities $\lambda_n, \lambda_m>0$, respectively.

Based on the implicit results of Corollaries 1 and 2 of this article and Theorem 1 of this article, I think we should be able to write $$dN_t dM_t = 0.$$

Can anyone please help me with the proof of this equation?

1 Answers1

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If $M$ and $N$ are independent (your references appear to make this assumption), then $M+N$ is also a Poisson process. So, using the polarization identity:

$$ dMdN = 2^{-1}\left[(d(M+N))^2 - (dM)^2 - (dN)^2\right] $$

$$ = 2^{-1}\left[d(M+N) - dM - dN \right] = 0 $$

(A proof of $(dX)^2 = dX$ for a Poisson process $X$ is available here.)

ir7
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    Great answer (+1)! It made me think: For a Brownian motion, we have $dWdt=0$, $dW^2=dt$ and $dW^3=0$. Are there analogues for Poisson processes? Your link suggests $dN^2=dN$ and $dN^3=dN$, etc.? What about cross-terms like $dNdt$ or $dWdN$ (if $W$ and $N$ are independent)? – Alex Jun 26 '21 at 18:30
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    @Alex Yes, $(dN)^3 = (dN)^2\cdot (dN) = dN\cdot dN = dN$. And, yes, based on the so-called "Ito multiplication table for Brownian motion and jumps" (aka "and Poisson process"), the two cross-terms you mentioned are 0. Make the cross-terms a SE Quant question if you are looking for some sort of proofs or proof references. – ir7 Jun 26 '21 at 18:48
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    @Alex This resource has the table, Table 20.1 (but not the cross-term proofs, I think). – ir7 Jun 26 '21 at 19:01
  • cool, thank you very much for the table. That's really interesting. Thank you! – Alex Jun 26 '21 at 19:15
  • @ir7 Great help, thank you. You mentioned this result holds when $N$ and $M$ are independent. Could you please guide me through the calculations when they're not? For example, if $N_t \sim \text{Poisson}(\lambda t)$ and $M_t \sim \text{Poisson}(\theta \lambda t)$. Thanks in advance. –  Jun 27 '21 at 12:26
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    @VultraUiolet Make it a SE Quant question, basically, how does one create correlated Poisson processes (common shock model comes to mind; just stating different constant intensities does not imply process dependence) and then how does one compute their quadratic covariation. – ir7 Jun 27 '21 at 19:29