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A note on positive supermartingales in ruin theory
Abstract
In this note, we present an elementary proof of Kolmogorov’s inequality for positive supermartingales and an application of
this inequality to the ruin problem for a class of surplus processes for which an adjustment coefficient need to exist.
Die vorliegende Arbeit enthält einen elementaren Beweis der Kolmogorovschen Ungleichung für positive Supermartingale sowie
eine Anwendung dieser Ungleichung auf das Ruin-Problem für eine Klasse von Reserve-Prozessen für die ein Anpassungskoeffizient
nicht notwendigerweise existiert.
Recently, Csörgö and Steinebach proposed to estimate the adjustment coefficient in risk theory via a quantile type estimate based upon a sequence of intermediate order statistics. In the present paper, further alternative estimators are discussed which may be viewed as convex combinations of a Hill type and a quantile type estimate. Consistency is proved and rates of convergence are studied. Some simulation results are presented to illustrate the finite sample behavior of the proposed estimators.
A sequence of intermediate order statistics is proposed to estimate the adjustment coefficient in risk theory. The underlying r.v.’s may be viewed as maximum waiting times in busy cycles of GI⧸G⧸1 queueing models under light traffic. We prove strong consistency and study also rates of convergence. We give an example, and present simulation studies as well, for illustrating the behaviour of the proposed estimator.
The paper presents an algorithm of bounding from above and from below the deficit distribution at ruin. The algorithm is based on iterating a monotone integral operator which has a fixed point at the deficit distribution at ruin. The upper and lower bounds converge monotonically with an exponential rate to the exact value of the deficit distribution. A counterpart of the famous Cramér-Lundberg inequality for the distribution of the deficit at ruin is established. A general method to improve further the algorithm is given.
Sumario: A review of random variables -- Stochastic processes -- Martingales -- The distribution of aggregate claims in one year -- Principles of premium calculation -- Credibility and experience rating -- Risk exchange and reinsurance -- Ruin theory -- Some decision theoretical problems
Necessary and sufficient conditions are given which make sure that the adjustment coefficient exists. By a class of counterexamples it is shown that the classical conditions by Wald (1947), which are sufficient if the moment generating function of the corresponding random variable is finite everywhere, do not work in general.