Abstract
In this paper, we introduce tail dependene measures for collateral losses from catastrophic events. To calculate these measures, we use bivariate compound process where a Cox process with shot noise intensity is used to count collateral losses. A homogeneous Poisson process is also examined as its counterpart for the case where the catastrophic loss frequency rate is deterministic. Joint Laplace transform of the distribution of the aggregate collateral losses is derived and joint Fast Fourier transform is used to obtain the joint distributions of aggregate collateral losses. For numerical illustrations, a member of Farlie-Gumbel-Morgenstern copula with exponential margins is used. The figures of the joint distributions of collateral losses, their contours and numerical calculations of risk measures are also provided.
Highlights
Over the recent years, numerous papers have looked at the modelling of dependence within an insurance portfolio or between insurance portfolios [1,2,3,4,5]
We have used bivariate compound process to model aggregate collateral losses arising from catastrophic events such as flood, storm, hail, bushfire and earthquake
For the number of collateral losses, a Cox process was used to accommodate the stochastic nature of their frequency rate in practice
Summary
Numerous papers have looked at the modelling of dependence within an insurance portfolio or between insurance portfolios [1,2,3,4,5]. The shot noise process is useful to loss arrival process as it measures the frequency, magnitude and time period needed to determine the effect of catastrophic events. The shot noise process can be used as the parameter of a Cox process to measure the number of catastrophic losses, i.e. we will use it as an intensity function to generate a Cox process. We present definitions and important properties L t2 with the aid of piecewise deterministic of L t1 Markov and processes theory [16,24,25] This theory is used to derive joint Laplace transform of the distribution of the aggregate collateral losses L t1 and L t2
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