In statistics and probability theory, a point process is a collection of mathematical points randomly located within a mathematical space such as the real line. In earthquake science, such processes are often used to model the occurrence of earthquake events in time, as stress builds in the Earth’s crust before being released at specific moments.
A sub-class of point processes of great interest are so-called self-correcting processes, for which the occurrence of past points inhibits the occurrence of future points.
In a recent paper, LML Fellow Jiancang Zhang and colleagues consider generalized models of the stress-release process based on a real-valued random variable. They refer to these as extrinsic stress-release processes, and derive a new formula for the distribution of the joint inter-arrival times. As a direct consequence, they also propose an exact simulation algorithm which, in comparison to standard methods, offers an alternative method for generating sample paths. The extension discussed is motivated by the influence of exogenous geophysical data on earthquake occurrence.
In their analysis, Zhang and colleagues offer a detailed comparison of the new exact algorithm to the standard thinning algorithm. In addition, they also present an infinitesimal generator for extrinsic stress-release processes, which is closely linked to the martingale problem used to characterize weak solutions of partial integro-differential equations. This approach allows the authors to derive the reciprocal moments of the intensity function, which they use to demonstrate the correctness of our simulation algorithms. The researchers hope these new algorithms can be used to produce more reliable predictions of damage due to a range of earthquake scenarios.
These results are summarised in the Journal of Applied Probability, 59(1), 2022, (forthcoming).
An arxiv version is available at https://arxiv.org/pdf/2106.