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Joint moments of discounted claims and discounted perturbation until ruin in the compound Poisson risk model with diffusion

Published online by Cambridge University Press:  31 March 2022

Eric C. K. Cheung Open the ORCID record for Eric C. K. Cheung [Opens in a new window]  and
Haibo Liu
Eric C. K. Cheung
Affiliation:
School of Risk and Actuarial Studies, UNSW Business School, University of New South Wales, Sydney, NSW 2052, Australia. E-mail: eric.cheung@unsw.edu.au
Haibo Liu
Affiliation:
Department of Statistics and Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907, USA
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Abstract

This paper studies a generalization of the Gerber-Shiu expected discounted penalty function [Gerber and Shiu (1998). On the time value of ruin. North American Actuarial Journal 2(1): 48–72] in the context of the perturbed compound Poisson insurance risk model, where the moments of the total discounted claims and the discounted small fluctuations (arising from the Brownian motion) until ruin are also included. In particular, the latter quantity is represented by a stochastic integral and has never been analyzed in the literature to the best of our knowledge. Recursive integro-differential equations satisfied by our generalized Gerber-Shiu function are derived, and these are transformed to defective renewal equations where the components are identified. Explicit solutions are given when the individual claim amounts are distributed as a combination of exponentials. Numerical illustrations are provided, including the computation of the covariance between discounted claims and discounted perturbation until ruin.

Keywords

Discounted claims until ruinDiscounted perturbationExpected discounted penalty functionInsurance risk modelJoint moments
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 37 , Special Issue 2: Probability and stochastic modeling in actuarial science and related fields , April 2023 , pp. 387 - 417
Copyright
© The Author(s), 2022. Published by Cambridge University Press

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