On a multi-threshold compound Poisson process perturbed by diffusion |
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Authors: | Ilie-Radu Mitric Kristina P Sendova Cary Chi-Liang Tsai |
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Institution: | 1. Department of Statistical and Actuarial Sciences, University of Western Ontario, London, ON, N6A 5B7, Canada;2. Department of Statistics and Actuarial Science, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada |
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Abstract: | We consider a multi-layer compound Poisson surplus process perturbed by diffusion and examine the behaviour of the Gerber–Shiu discounted penalty function. We derive the general solution to a certain second order integro-differential equation. This permits us to provide explicit expressions for the Gerber–Shiu function depending on the current surplus level. The advantage of our proposed approach is that if the diffusion term converges to zero, the above-mentioned explicit expressions converge to those under the classical compound Poisson model, provided that the same initial conditions apply. This is subsequently illustrated by an extended example related to the probability of ultimate ruin. |
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