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.
A. de Acosta, “Exponential Tightness and Projective Systems in Large Deviation Theory,” In: D. Pollard, E. Togersen and G. Yang, Eds., Festschrift for Lucien Le Cam, Springer, New York, 1997, pp. 143-156.
R. L. Dobrushin and E. A. Pechersky, “Large Deviations for Tandem Queueing Systems,” Journal of Applied Ma- thematics and Stochastic Analysis, Vol. 7, No. 3, 1994, pp. 301-330. doi:10.1155/S1048953394000274
A. Ganesh, C. Macci and G. L.Torrisi, “A Class of Risk Processes with Reserve-Dependent Premium Rate: Sample Path Large Deviations and Importance Sampling,” Queueing Systems, Vol. 55, No. 2, 2007, pp. 83-94.
A. Ganesh, C. Macci and G. L. Torrisi, “Sample Path Large Deviations Principles for Poisson Shot Noise Proc- esses, and Applications,” Electronic Journal of Probabil- ity, Vol. 10, No. 32, 2005, pp. 1026-1043.
S. Asmussen and C. Klüppelberg, “Large Deviations Results for Subexponential Tails, with Applications to Insurance Risk,” Stochastic Processes and their Applications, Vol. 64, No. 1, 1996, pp. 103-125.
C. Klüppelberg and T. Mikosch, “Large Deviations of Heavy-Tailed Random Sums with Applications in Insurance and Finance,” Journal of Applied Probability, Vol. 34, No. 2, 1997, pp. 293-308. doi:10.2307/3215371
G. R. Shorack and J. A. Wellner, “Empirical processes with applications to statistics of Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics,” John Wiley & Sons Inc., New York, 1986.