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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

Part of: Stochastic processes Mathematical economics Markov processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Angelos Dassios  and
Ji-Wook Jang
Angelos Dassios*
Affiliation:
London School of Economics
Ji-Wook Jang*
Affiliation:
University of New South Wales
*
∗Postal address: Department of Statistics, London School of Economics, Houghton Street, London WC2A 2AE, UK. Email address: a.dassios@lse.ac.uk
∗∗Postal address: Actuarial Studies, Faculty of Commerce and Economics, University of New South Wales, Sydney, NSW 2052, Australia. Email address: j.jang@unsw.edu.au
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Abstract

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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.

Keywords

Kalman-Bucy filterGaussian processCox processshot noise processpiecewise-deterministic Markov process theorystop-loss reinsurance contract

MSC classification

Secondary: 60G35: Signal detection and filtering 60F05: Central limit and other weak theorems 60G55: Point processes 60J75: Jump processes 91B30: Risk theory, insurance
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 1 , March 2005 , pp. 93 - 107
Copyright
© Applied Probability Trust 2005 

Footnotes

Ji-Wook Jang acknowledges the scholarship awarded by the Association of British Insurers for this research.

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