Why Independent Poisson Processes Have Zero Cross Variation

Summary

The document explains why the product of infinitesimal increments of two independent Poisson processes is zero in Itô calculus. It applies the polarization identity to express the cross product in terms of squared increments of each process and of their sum.

For a Poisson process, the squared increment equals the increment in the differential calculus used here. Independence ensures that the sum is also a Poisson process, so substituting this property into the identity leaves a zero cross term. This argument depends on independence; the document offers no derivation for dependent processes or processes that may jump simultaneously. It is a brief proof sketch and points to separate material for the squared-increment property.

Key ideas

• The polarization identity rewrites the product of two increments using squared increments.
• A Poisson process has a squared differential equal to its differential in the stated calculus.
• The sum of independent Poisson processes is Poisson, which makes the cross differential vanish.
• The conclusion relies on independence and does not address processes with shared jumps.

Tags

# Ito multiplication


# Ito multiplication












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?

## Answer by ir7 (score 7, accepted)

https://quant.stackexchange.com/a/65729

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.)

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.