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Ruin probabilities via local adjustment coefficients

Part of: Probabilistic methods, simulation and stochastic differential equations Limit theorems Special processes

Published online by Cambridge University Press:  14 July 2016

Søren Asmussen  and
Hanne Mandrup Nielsen
Søren Asmussen*
Affiliation:
Aalborg University
Hanne Mandrup Nielsen*
Affiliation:
Baltica Insurance Company, Copenhagen
*
∗Postal address: Institute of Electronic Systems, Aalborg University, Fr. Bajersv. 7E, DK 9220 Aalborg, Denmark.
∗∗Postal address: Baltica Insurance Company, Copenhagen, Denmark.
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Abstract

Let ψ(u) be the ruin probability in a risk process with initial reserve u, Poisson arrival rate β, claim size distribution B and premium rate p(x) at level x of the reserve. Let y(x) be the non-zero solution of the local Lundberg equation . It is shown that is non-decreasing and that log ψ(u) ≈ –I(u) in a slow Markov walk limit. Though the results and conditions are of large deviations type, the proofs are elementary and utilize piecewise comparisons with standard risk processes with a constant p. Also simulation via importance sampling using local exponential change of measure defined in terms of the γ(x) is discussed and some numerical results are presented.

Keywords

FAST SIMULATIONLARGE DEVIATIONSLUNDBERG INEQUALITYMARTINGALERARE EVENTSSLOW MARKOV WALKSTORAGE PROCESS

MSC classification

Primary: 60F10: Large deviations
Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.) 65C05: Monte Carlo methods
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 3 , September 1995 , pp. 736 - 755
Copyright
Copyright © Applied Probability Trust 1995 

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