Liu, Shiju
(2017)
*Excursions of risk processes with inverse gaussian processes and their applications in insurance.*
PhD thesis, The London School of Economics and Political Science (LSE).

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## Abstract

Parisian excursion of a Levy process is defined as the excursion of the process below or above a pre-defined barrier continuously exceeding a certain time length. In this thesis, we study classical and Parisian type of ruin problems, as well as Parisian excursions of collective risk processes generalized on the classical Cramer-Lundberg risk model. We consider that claim sizes follow mixed exponential distributions and that the main focus is claim arrival process converging to an inverse Gaussian process. By this convergence, there are infinitely many and arbitrarily small claim sizes over any finite time interval. The results are obtained through Gerber-Shiu penalty function employed in an infinitesimal generator and inverting corresponding Laplace transform applied to the generator. In Chapter 3, the classical collective risk process under the Cram´er-Lundberg risk model framework is introduced, and probabilities of ruin with claim sizes following exponential distribution and a combination of exponential distributions are also studied. In Chapter 4, we focus on a surplus process with the total claim process converging to an inverse Gaussian process. The classical probability of ruin and the joint distribution of ruin time, overshoot and initial capital are given. This joint distribution could provide us with probabilities of ruin given different initial capitals in any finite time horizon. In Chapter 5, the classical ruin problem is extended to Parisian type of ruin, which requires that the length of excursions of the surplus process continuously below zero reach a predetermined time length. The joint law of the first excursion above zero and the first excursion under zero is studied. Based on the result, the Laplace transform of Parisian ruin time and formulae of probability of Parisian type of ruin with different initial capitals are obtained. Considering the asymptotic properties of claim arrival process, we also propose an approximation of the probability of Parisian type of ruin when the initial capital converges to infinity. In Chapter 6, we generalize the surplus process to two cases with total claim process still following an inverse Gaussian process. The first generalization is the case of variable premium income, in which the insurance company invests previous surplus and collects interest. The probability of survival and numerical results are given. The second generalization is the case in which capital inflow is also modelled by a stochastic process, i.e. a compound Poisson process. The explicit formula of the probability of ruin is provided.

Item Type: | Thesis (PhD) |
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Additional Information: | © 2017 Shiju Liu |

Library of Congress subject classification: | H Social Sciences > HA Statistics |

Sets: | Departments > Statistics |

Supervisor: | Dassios, Angelos |

URI: | http://etheses.lse.ac.uk/id/eprint/3637 |

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