Applications of Mogulskii, and Kurtz-Feng Large Deviation Results to Risk Reserve Processes with Aggregate Claims

Jorge Garcia, Ana Meda

Departamento de Matemáticas, Facultad de Ciencias, Mexico City, Mexico.
Department of Mathematics, California State University Channel Islands, Camarillo, USA.

DOI: 10.4236/am.2012.312A291                

Abstract

In this paper we examine the large deviations principle (LDP) for sequences of classic Cramér-Lundberg risk processes under suitable time and scale modifications, and also for a wide class of claim distributions including (the non-super- exponential) exponential claims. We prove two large deviations principles: first, we obtain the LDP for risk processes on D∈[0,1] with the Skorohod topology. In this case, we provide an explicit form for the rate function, in which the safety loading condition appears naturally. The second theorem allows us to obtain the LDP for Aggregate Claims processes on D∈[0,∞) with a different time-scale modification. As an application of the first result we estimate the ruin probability, and for the second result we work explicit calculations for the case of exponential claims.

Keywords

Large Deviations; Cramer-Lundberg Reserve Risk Processes; Probability Theory and Mathematical, Statistics in Insurance; Stochastic Models for Claim Frequency; Claim Size and Aggregate Claims;Reserves

Conflicts of Interest

The authors declare no conflicts of interest.

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