Catastrophe Insurance Modeled by Shot-Noise Processes

Abstract

Shot-noise processes generalize compound Poisson processes in the following way: a jump (the shot) is followed by a decline (noise). This constitutes a useful model for insurance claims in many circumstances; claims due to natural disasters or self-exciting processes exhibit similar features. We give a general account of shot-noise processes with time-inhomogeneous drivers inspired by recent results in credit risk. Moreover, we derive a number of useful results for modeling and pricing with shot-noise processes. Besides this, we obtain some highly tractable examples and constitute a useful modeling tool for dynamic claims processes. The results can in particular be used for pricing Catastrophe Bonds (CAT bonds), a traded risk-linked security. Additionally, current results regarding the estimation of shot-noise processes are reviewed.

Moments of renewal shot-noise processes and their applications

“…for g(z, s, u, t) to belong to the domain of the (extended) generator A. For the details on finding the generator of (Z t , S t , U t , t) using the piecewise deterministic Markov process theory (Davis, 1984(Davis, , 1993, see Dassios and Embrechts (1989), Dassios and Jang (2003), Rolski et al (2008), Dassios and Zhao (2011, 2014) and many others.…”

“…Since the beginning of the 20th century, shot-noise processes have been extensively used to model a very wide variety of natural phenomena, with numerous applications in electronics, optics, biology and many other fields in natural science, see early literature in Campbell (1909a,b), Schottky (1918), Picinbono et al (1970) and Verveen and DeFelice (1974). More recent applications extended to insurance and actuarial science in particular can be found in Klüppelberg and Mikosch (1995), Brémaud (2000), Jang (2003, 2005), Jang (2004), Jang and Krvavych (2004), Torrisi (2004), Albrecher and Asmussen (2006), Macci and Torrisi (2011), Zhu (2013) and Schmidt (2014). Mostly, they adopted the classical Poisson shot-noise process (Cox and Isham, 1980, p.88),…”

A micro-level claim count model with overdispersion and reporting delays

“…Suppose that Ω(t) is a continuously differentiable function with dΩ(t) = Ω t (t) dt. One should notice that Λ (t) is a semi-martingale (Schmidt, 2014) and by using the Itō formula for semi-martingales (Proposition 8.19, Cont and Tankov, 2004) we have…”

A risk model with renewal shot-noise Cox process

“…Therefore, the shot-noise process can be used as the intensity of a Cox process to measure the number of catastrophic losses. Previous works on insurance applications using a shot-noise process or a Cox process with shot-noise intensity can be found in Klüppelberg and Mikosch (1995), Brémaud (2000), Dassios and Jang (2003), Jang and Krvavych (2004), Torrisi (2004), Dassios and Jang (2005), Albrecher and Asmussen (2006), Macci and Torrisi (2011), Zhu (2013) and Schmidt (2014).…”