Shared-Shock Models for Correlated Poisson Processes

Summary

The document explains one way to model dependence between two Poisson processes: construct each from its own independent arrivals plus a shared Poisson process. If the processes are M = X + Z and N = Y + Z, the shared component Z creates simultaneous jumps. The covariance of their counts comes from that common component, and the document gives a formula for their correlation in terms of the component intensities.

Using differential notation, products of increments from independent processes vanish, while a Poisson increment squared equals that increment. Expanding the product dM dN therefore leaves dZ, representing the shared jumps. This is a special common-shock construction, not a formula for arbitrary correlated Poisson processes. The original question's idea of treating the sum as a Poisson process does not apply here, since the shared jumps are counted twice in M + N. The example offers a useful way to reason about correlated event arrivals, but does not develop a general dependence model or trading application.

Key ideas

• A common-shock construction makes two Poisson processes dependent by including a shared Poisson component in each.
• The covariance between the processes' counts is generated by the intensity of their shared component.
• The product of their increments equals the shared process increment under this construction.
• The sum of the two constructed processes is not generally Poisson because shared jumps are counted twice.

Tags

# Covariation of these processes


# Covariation of these processes












Let $N_t \sim \text{Poisson}(\lambda t)$ and $M_t \sim \text{Poisson}(\theta \lambda t)$.

We know that if $N$ and $M$ were independent, $dNdM = 0$ using polarization identity. We also know that $(dN)^2 = dN$; but now that these two processes are correlated, how can we calculate $dNdM$ ?

I though about polarization identity and putting it in differential notations and given that $N+M$ is also a Poisson process, we can write \begin{align*} dNdM &= \frac{1}{2}\left[ \left(d(N+M)\right)^2 - (dN)^2 - (dM)^2 \right] \\ &= \frac{1}{2}\left[ d(N+M) - dN - dM \right] \end{align*} But how can we calculate $d(N+M)$?

## Answer by ir7 (score 3, accepted)

https://quant.stackexchange.com/a/65755

(Special case only.)

One special way to create correlated Poisson processes is using a common 'shock' model idea.

For $X$, $Y$, and $Z$ independent Poisson processes, let's define:

$$ M = X+ Z, \; \; N = Y+Z.$$

We note that $M$ and $N$ are Poisson processes, but that $M+N$ is not ($2Z$ is not a Poisson process).

We also note that the Pearson correlation between $M_t$ and $N_t$ is not time-dependent and it is always positive (since intensities are positive):

$$ \rho(M_t, N_t) = \frac{\lambda_Z}{\sqrt{(\lambda_X+\lambda_Z)((\lambda_Y+\lambda_Z)}} $$

Formally we also get:

$$ dMdN = (dX +dZ)(dY+dZ) = dXdY+dXdZ +dYdZ + (dZ)^2 = dZ. $$

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