Abstract
A statistical inference for ruin probability from a certain discrete sample of the surplus is discussed under a spectrally negative Lévy insurance risk. We consider the Laguerre series expansion of ruin probability, and provide an estimator for any of its partial sums by computing the coefficients of the expansion. We show that the proposed estimator is asymptotically normal and consistent with the optimal rate of convergence and estimable asymptotic variance. This estimator enables not only a point estimation of ruin probability but also an approximated interval estimation and testing hypothesis.
Highlights
Ruin probability has been one of the central topics for long time in insurance mathematics since the paper by Lundberg (1903), where a compound Poisson type surplus was supposed
Various stochastic surplus models have been considered, and we found that Lévy processes seem to be good candidates for insurance surplus models from several aspects: (1) computational convinience; (2) compatibility with financial theories and dynamical risk managements; (3) statistical prediction of the future surplus
We consider the statistical inference for ruin probability of Lévy insurance surplus under a certain sampling scheme
Summary
Ruin probability has been one of the central topics for long time in insurance mathematics since the paper by Lundberg (1903), where a compound Poisson type surplus was supposed. (2) compatibility with financial theories and dynamical risk managements; (3) statistical prediction of the future surplus. On the aspect (1): Lévy process has properties of independent and stationary increments, and it derives many beautiful mathematical formulae for ruin probability and other ruin-related quantities via the fluctuation theory of Lévy processes; see Huzak et al (2004), Feng and Shimizu (2013), and Kyprianou (2014), among others. On (2), Trufin et al (2011), Shimizu and Tanaka (2018) proposed dynamic risk measures based on ruin probability and its related quantities, which are useful in insurance and financial mathematics. We focus on the aspect (3), which is the most important step to make the ruin theory applicable in practice
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