Measuring Tail Dependence for Aggregate Collateral Losses Using Bivariate Compound Shot-Noise Cox Process

Abstract The Poisson process is an essential building block to move up to complicated counting processes, such as the Cox (“doubly stochastic Poisson”) process, the Hawkes (“self-exciting”) process, exponentially decaying shot-noise Poisson (simply “shot-noise Poisson”) process and the dynamic contagion process. The Cox process provides flexibility by letting the intensity not only depending on time but also allowing it to be a stochastic process. The Hawkes process has self-exciting property and clustering effects. Shot-noise Poisson process is an extension of the Poisson process, where it is capable of displaying the frequency, magnitude and time period needed to determine the effect of points. The dynamic contagion process is a point process, where its intensity generalises the Hawkes process and Cox process with exponentially decaying shot-noise intensity. To facilitate the usage of these processes in practice, we revisit the distributional properties of the Poisson, Cox, Hawkes, shot-noise Poisson and dynamic contagion process and their compound processes. We provide simulation algorithms for these processes, which would be useful to statistical analysis, further business applications and research. As an application of the compound processes, numerical comparisons of value-at-risk and tail conditional expectation are made.

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Pricing of catastrophe reinsurance and derivatives using the Cox process with shot noise intensity

Finance and Stochastics ◽

10.1007/s007800200079 ◽

2003 ◽

Vol 7 (1) ◽

pp. 73-95 ◽

Cited By ~ 122

Author(s):

Angelos Dassios ◽

Ji-Wook Jang

Keyword(s):

Shot Noise ◽

Noise Intensity ◽

Cox Process ◽

Catastrophe Reinsurance

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Kalman-Bucy Filtering for Linear Systems Driven by the Cox Process with Shot Noise Intensity and Its Application to the Pricing of Reinsurance Contracts

Journal of Applied Probability ◽

10.1017/s0021900200000085 ◽

2005 ◽

Vol 42 (01) ◽

pp. 93-107 ◽

Cited By ~ 4

Author(s):

Angelos Dassios ◽

Ji-Wook Jang

Keyword(s):

Gaussian Process ◽

Shot Noise ◽

Three Dimensional ◽

Arrival Rate ◽

Cox Process ◽

Noise Process ◽

Shot Noise Process ◽

Convergence Results ◽

Insurance Portfolio ◽

Aggregate Loss

In practical situations, we observe the number of claims to an insurance portfolio but not the claim intensity. It is therefore of interest to try to solve the ‘filtering problem’; that is, to obtain the best estimate of the claim intensity on the basis of reported claims. In order to use the Kalman-Bucy filter, based on the Cox process incorporating a shot noise process as claim intensity, we need to approximate it by a Gaussian process. We demonstrate that, if the primary-event arrival rate of the shot noise process is reasonably large, we can then approximate the intensity, claim arrival, and aggregate loss processes by a three-dimensional Gaussian process. We establish weak-convergence results. We then use the Kalman-Bucy filter and we obtain the price of reinsurance contracts involving high-frequency events.

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Kalman-Bucy Filtering for Linear Systems Driven by the Cox Process with Shot Noise Intensity and Its Application to the Pricing of Reinsurance Contracts

Journal of Applied Probability ◽

10.1239/jap/1110381373 ◽

2005 ◽

Vol 42 (1) ◽

pp. 93-107 ◽

Cited By ~ 16

Author(s):

Angelos Dassios ◽

Ji-Wook Jang

Keyword(s):

Gaussian Process ◽

Shot Noise ◽

Three Dimensional ◽

Arrival Rate ◽

Cox Process ◽

Noise Process ◽

Shot Noise Process ◽

Convergence Results ◽

Insurance Portfolio ◽

Aggregate Loss

In practical situations, we observe the number of claims to an insurance portfolio but not the claim intensity. It is therefore of interest to try to solve the ‘filtering problem’; that is, to obtain the best estimate of the claim intensity on the basis of reported claims. In order to use the Kalman-Bucy filter, based on the Cox process incorporating a shot noise process as claim intensity, we need to approximate it by a Gaussian process. We demonstrate that, if the primary-event arrival rate of the shot noise process is reasonably large, we can then approximate the intensity, claim arrival, and aggregate loss processes by a three-dimensional Gaussian process. We establish weak-convergence results. We then use the Kalman-Bucy filter and we obtain the price of reinsurance contracts involving high-frequency events.

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The Distribution of the Interval between Events of a Cox Process with Shot Noise Intensity

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/2008/367170 ◽

2008 ◽

Vol 2008 ◽

pp. 1-14 ◽

Cited By ~ 5

Author(s):

Angelos Dassios ◽

Jiwook Jang

Keyword(s):

Laplace Transform ◽

Generating Function ◽

Shot Noise ◽

Noise Intensity ◽

Probability Generating Function ◽

Cox Process ◽

Noise Process ◽

Shot Noise Process ◽

Piecewise Deterministic Markov Processes ◽

The Laplace Transform

Applying piecewise deterministic Markov processes theory, the probability generating function of a Cox process, incorporating with shot noise process as the claim intensity, is obtained. We also derive the Laplace transform of the distribution of the shot noise process at claim jump times, using stationary assumption of the shot noise process at any times. Based on this Laplace transform and from the probability generating function of a Cox process with shot noise intensity, we obtain the distribution of the interval of a Cox process with shot noise intensity for insurance claims and its moments, that is, mean and variance.

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