This document discusses using bivariate counting processes to model risk from correlated catastrophic events like earthquakes. It provides motivation for why correlated risks are important from examples of large earthquakes triggering aftershocks and increased seismic activity near the main shock. Specifically, it examines using bivariate counting processes to model the number of earthquakes before and after a major earthquake, which shows a significant increase in seismic activity for a few rupture lengths around the main shock.

1. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management
Bivariate Counting Processes
for Risk Management
Mathieu Boudreault & Arthur Charpentier
Université du Québec à Montréal
charpentier.arthur@uqam.ca
http ://freakonometrics.blog.free.fr/
IFM2, Mathematical Finance Days, May 2012
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2. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management
Figure : Mexican catastrophe bond, 2006-2009, via Cabrera (2006)
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3. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management
Motivation : Mexican (earthquake) catastrophe bond
Figure : Mexican catastrophe bond, 2006-2009, via Cabrera (2006)
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4. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management
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5. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management
Motivation
Figure : Time and distance distribution (to 6,000 km) of large (5<M<7)
aftershocks from 205 M≥7 mainshocks (in sec. and h.). Parsons & Velasco
(2011)
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6. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management
Motivation
“Large earthquakes are known to trigger earthquakes elsewhere. Damaging large
aftershocks occur close to the mainshock and microearthquakes are triggered by
passing seismic waves at significant distances from the mainshock. It is unclear,
however, whether bigger, more damaging earthquakes are routinely triggered at
distances far from the mainshock, heightening the global seismic hazard after
every large earthquake. Here we assemble a catalogue of all possible earthquakes
greater than M5 that might have been triggered by every M7 or larger mainshock
during the past 30 years. [...] We observe a significant increase in the rate of
seismic activity at distances confined to within two to three rupture lengths of the
mainshock. Thus, we conclude that the regional hazard of larger earthquakes is
increased after a mainshock, but the global hazard is not.” Parsons & Velasco
(2011)
Figure : Number of earthquakes (magnitude exceeding 2.0, per 15 sec.) following
a large earthquake (of magnitude 6.5), normalized so that the expected number
of earthquakes before and after is 100.
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7. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management
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