Martingale Approach for Moments of Discounted Aggregate Claims

Jiwook Jang

47 citations0/0 files storedJournal of Risk & Insurance

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Probability and Risk Models Probability and Statistical Research Insurance, Mortality, Demography, Risk Management

Abstract

Abstract We examine the Laplace transform of the distribution of the shot noise process using the martingale. Applying the piecewise deterministic Markov processes theory and using the relationship between the shot noise process and the accumulated/discounted aggregate claims process, the Laplace transform of the distribution of the accumulated aggregate claims is obtained. Assuming that the claim arrival process follows the Poisson process and claim sizes are assumed to be exponential and mixture of exponential, we derive the explicit expressions of the actuarial net premiums and variances of the discounted aggregate claims, which are the annuities paid continuously. Numerical examples are also provided based on them.

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References

1984Piecewise-Deterministic Markov Processes: A General Class of Non-Diffusion Stochastic Models 1979An introduction to mathematical risk theory 2001Stochastic Processes for Insurance and Finance 1970Mathematical Methods in Risk Theory 1971Mathematical Methods in Risk Theory.1990The Distribution of a Perpetuity, with Applications to Risk Theory and Pension Funding 2000Stochastic Processes for Insurance and Finance 2003Pricing of catastrophe reinsurance and derivatives using the Cox process with shot noise intensity

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