Μελέτη του χρόνου χρεοκοπίας και σχετικών μέτρων χρεοκοπίας
Study of the distribution of the time to ruin and related ruin measures
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Χρόνος χρεοκοπίας ; Σχετικά μέτρα χρεοκοπίας ; Συνάρτηση Gerber-Shiu ; Μοντέλο κινδύνου Sparre AndersenAbstract
Following the introduction of the discounted penalty function by Gerber and Shiu (1998),
significant progress has been made on the analysis of various ruin-related quantities in risk
theory. As we know, the discounted penalty function not only provides a systematic platform to
jointly analyze various quantities of interest, but also offers the convenience to extract key pieces
of information from a risk management prospective. For example, by eliminating the penalty
function, the Gerber-Shiu function becomes the Laplace-Stieltjes transform of the time to ruin,
inversion of which results in a series expansion for the associated density of the time to ruin (see,
e.g., Dickson and Willmot (2005)). In this thesis, we propose to analyze the long-standing finite time ruin problem by incorporating the number of claims until ruin into the Gerber-Shiu analysis.
As will be seen in Chapter 2, many nice analytic properties of the original Gerber-Shiu function
are preserved by this generalized analytic tool. For instance, the Gerber-Shiu function still
satisfies a defective renewal equation and can be generally expressed in terms of some roots of
Lundberg’s generalized equation in the Sparre Andersen risk model.
In this thesis, we propose not only to unify previous methodologies on the study of the
density of the time to ruin through the use of Lagrange’s expansion theorem, but also to provide
insight into the nature of the series expansion by identifying the probabilistic contribution of
each term in the expansion through analysis involving the distribution of number of claims until
ruin. In Chapter 3, we study the joint generalized density of the time to ruin and the number of
claims until ruin in the classical compound Poisson risk model. We also utilize an alternative
approach to obtain the density of the time to ruin based on the Lagrange inversion technique
introduced by Dickson and Willmot (2005). In Chapter 4, relying on the Lagrange expansion
theorem for analytic inversion, the joint density of the time to ruin, the surplus immediately
before ruin and the number of claims until ruin is examined in the Sparre Andersen risk model
with exponential claim sizes and arbitrary interclaim times.
To our knowledge, existing results on the finite-time ruin problem in the Sparre Andresen risk
model typically involve an exponential assumption on either the interclaim times or the claim
sizes (see, e.g., Borovkov and Dickson (2008)). Among the few exceptions, we mention Dickson
and Li (2010, 2012) who analyzed the density of the time to ruin for Erlang-𝑛 interclaim times.
In Chapter 5, we propose a significant breakthrough by utilizing the multivariate version of
Lagrange’s expansion theorem to obtain a series expansion for the density of the time to ruin
under a more general distribution assumption, namely when interclaim times are distributed as a
combination of 𝑛 exponentials. It is worth emphasizing that this technique can also be applied to
other areas of applied probability. For instance, the proposed methodology can used to obtain the
distribution of some first passage times for particular stochastic processes. As an illustration, the
duration of a busy period in a queueing risk model will be examined.
Finally, some concluding remarks and discussion of future research are made in Chapter 6.
Postgraduate Studies Programme
Αναλογιστική Επιστήμη και Διοικητική ΚινδύνουDepartment
Σχολή Χρηματοοικονομικής και Στατιστικής. Τμήμα Στατιστικής και Ασφαλιστικής ΕπιστήμηςNumber of pages
112Language
GreekCollections
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Αναφορά Δημιουργού-Μη Εμπορική Χρήση-Όχι Παράγωγα Έργα 3.0 Ελλάδα
Αναφορά Δημιουργού-Μη Εμπορική Χρήση-Όχι Παράγωγα Έργα 3.0 Ελλάδα