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

Affiliation(s)

Department of Mathematics, California State University Channel Islands, Camarillo, USA.

Departamento de Matemáticas, Facultad de Ciencias, Mexico City, Mexico.

Department of Mathematics, California State University Channel Islands, Camarillo, USA.

Departamento de Matemáticas, Facultad de Ciencias, Mexico City, Mexico.

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.

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

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

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

Cite this paper

J. Garcia and A. Meda, "Applications of Mogulskii, and Kurtz-Feng Large Deviation Results to Risk Reserve Processes with Aggregate Claims,"*Applied Mathematics*, Vol. 3 No. 12, 2012, pp. 2109-2117. doi: 10.4236/am.2012.312A291.

J. Garcia and A. Meda, "Applications of Mogulskii, and Kurtz-Feng Large Deviation Results to Risk Reserve Processes with Aggregate Claims,"

References

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[2] J. Lynch and J. Sethuraman, “Large Deviations for Processes with Independent Increments,” Annals of Probability, Vol. 15, No. 2, 1987, pp. 610-627. doi:10.1214/aop/1176992161

[3] A. de Acosta, “Large Deviations for Vector-Valued Lévy Processes,” Stochastic Processes and Their Applications, Vol. 51, No. 1, 1994, pp. 75-115. doi:10.1016/0304-4149(94)90020-5

[4] 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.

[5] Z. H. Li and E. A. Pechersky, “On Large Deviations in Queuing Systems,” Resenha, Vol. 4, No. 2, 1999, pp. 163-182.

[6] 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

[7] 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. doi:10.1007/s11134-006-9000-y

[8] 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.

[9] J. Feng and T. G. Kurtz, “Large Deviations for Stochastic Processes,” Vol. 131, Mathematical Surveys and Monographs, American Mathematical Society, Providence, 2006.

[10] S. Asmussen, “Ruin Probabilities,” Vol. 2, Advanced Series on Statistical Science & Applied Probability, World Scientific Publishing Co. Inc., River Edge, 2000.

[11] 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. doi:10.1016/S0304-4149(96)00087-7

[12] 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

[13] 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.

[14] P. Embrechts, C. Klüppelberg and T. Mikosch, “Modelling Extremal Events,” Vol. 33, Applications of Mathematics, Springer-Verlag, Berlin, 1997.

[1] A. A. Mogulskii, “Large Deviations for Processes with Independent Increments,” Annals of Probability, Vol. 21, No. 1, 1993, pp. 202-215. doi:10.1214/aop/1176989401

[2] J. Lynch and J. Sethuraman, “Large Deviations for Processes with Independent Increments,” Annals of Probability, Vol. 15, No. 2, 1987, pp. 610-627. doi:10.1214/aop/1176992161

[3] A. de Acosta, “Large Deviations for Vector-Valued Lévy Processes,” Stochastic Processes and Their Applications, Vol. 51, No. 1, 1994, pp. 75-115. doi:10.1016/0304-4149(94)90020-5

[4] 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.

[5] Z. H. Li and E. A. Pechersky, “On Large Deviations in Queuing Systems,” Resenha, Vol. 4, No. 2, 1999, pp. 163-182.

[6] 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

[7] 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. doi:10.1007/s11134-006-9000-y

[8] 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.

[9] J. Feng and T. G. Kurtz, “Large Deviations for Stochastic Processes,” Vol. 131, Mathematical Surveys and Monographs, American Mathematical Society, Providence, 2006.

[10] S. Asmussen, “Ruin Probabilities,” Vol. 2, Advanced Series on Statistical Science & Applied Probability, World Scientific Publishing Co. Inc., River Edge, 2000.

[11] 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. doi:10.1016/S0304-4149(96)00087-7

[12] 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

[13] 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.

[14] P. Embrechts, C. Klüppelberg and T. Mikosch, “Modelling Extremal Events,” Vol. 33, Applications of Mathematics, Springer-Verlag, Berlin, 1997.