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Martingale Approach for Moments of Discounted Aggregate Claims
Author(s) -
Jang JiWook
Publication year - 2004
Publication title -
journal of risk and insurance
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.055
H-Index - 63
eISSN - 1539-6975
pISSN - 0022-4367
DOI - 10.1111/j.0022-4367.2004.00086.x
Subject(s) - laplace transform , mathematics , martingale (probability theory) , econometrics , laplace distribution , aggregate (composite) , exponential function , poisson distribution , mathematical economics , statistics , mathematical analysis , materials science , composite material
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.