Discrete Time Homogeneous Markov Processes for the Study of the Basic Risk Processes

In this paper Markov models useful for following the time evolution of the aggregate claim amount and the claim number in the homogeneous time environment are presented. More precisely the homogeneous Markov reward processes in both discounted and not discounted cases are applied to solve the aggregate claim amount and the claim number processes respectively. In the last section the application of the proposed models is presented. Two different real-world databases are mixed for the construction of input data.

[1]  F. Riesz,et al.  Les systèmes d'équations linéaires : a une infinité d'inconnues , 1952 .

[2]  An algebraic treatment of fluctuations of sums of random variables , 1967 .

[3]  History of Actuarial Science , 1995 .

[4]  Tak Kuen Siu,et al.  Markov Chains: Models, Algorithms and Applications , 2006 .

[5]  Jac J. Janssen,et al.  Probabilités de Ruine pour une Classe de Modèles de Risque Semi-Markoviens , 1985, ASTIN Bulletin.

[6]  Jacques Janssen,et al.  Applied Semi-Markov Processes , 2005 .

[7]  D. Hernández-Hernández,et al.  A characterization of the optimal risk-sensitive average cost in finite controlled Markov chains , 2005, math/0503478.

[8]  P. Embrechts,et al.  Quantitative Risk Management: Concepts, Techniques, and Tools , 2005 .

[9]  S. Asmussen.  Risk theory in a Markovian environment , 1989 .

[10]  J. Janssen,et al.  Mathematical Finance: Deterministic and Stochastic Models , 2009 .

[11]  William H. Sanders,et al.  Reward Model Solution Methods with Impulse and Rate Rewards: An Algorithm and Numerical Results , 1994, Perform. Evaluation.

[12]  Some Transient Results on the M/SM/1 Special Semi-Markov Model in Risk and Queueing Theories , 1980, ASTIN Bulletin.

[13]  J. Janssen,et al.  Semi-Markov Risk Models for Finance, Insurance and Reliability , 2007 .

[14]  Thomas Mikosch,et al.  Non-Life Insurance Mathematics: An Introduction with the Poisson Process , 2006 .

[15]  R. D. Carmichael.  Review: Frédéric Riesz, Les Systèmes d'Équations linéaires a une Infinité d'Inconnues , 1914 .

[16]  Sean P. Meyn,et al.  Risk-Sensitive Optimal Control for Markov Decision Processes with Monotone Cost , 2002, Math. Oper. Res..

[17]  Harald Cramér,et al.  Collective risk theory , 1955 .

[18]  Tomasz Rolski,et al.  Stochastic Processes for Insurance and Finance , 2001 .

[19]  T. Rolski.  Stochastic Processes for Insurance and Finance , 1999 .

[20]  H. Cramér.  On the Mathematical Theory of Risk , 1994 .

[21]  J. Reinhard,et al.  On a Class of Semi-Markov Risk Models Obtained as Classical Risk Models in a Markovian Environment , 1984, ASTIN Bulletin.

[22]  R. Howard,et al.  Risk-Sensitive Markov Decision Processes , 1972 .