Matrix analytic methods for computations in risk theory

Abstract

The introduction of matrix analytic methods in risk theory has marked a significant progress in computations in risk theory. Matrix analytic methods have proven to be powerful computational tools for numerically analyzing complex risk models that traditional methods often had difficulty with. This is particularly noteworthy in the modern age of advanced computing and big data. Moving away from the traditional view of collective risk theory, we can now consider risk models that comprise of many stochastic processes of which data are abundant. These models may fall under the existing class of risk models; however, these more realistic risk models involve a large number of variables which increases the computational complexity significantly. Matrix analytic methods can provide reliable computing algorithms for risk models of such computational complexity, which have not been numerically feasible to analyze with the traditional computational tools in risk theory. This thesis is dedicated to improving the accessibility of the matrix analytic methodology in risk theory and developing further generalizations of the existing matrix analytic methods in risk theory in the attempt to promote its computational use. Although the literature of matrix analytic methods in risk theory is in its early stage, it is believed that the advancement in computations in risk theory brought by the matrix analytic methods will broaden the spectrum of problems in the risk theory literature in the direction of more realistic and practical risk models and computational analyses of these models. This will make risk theory as a whole more appealing to practitioners and those who are looking for more advanced actuarial risk management tools.