Information Entropy Method In Actuarial Science

Survival analysis and risk theory are two central problems in actuarial science. Based on the informational entropy theory, this dissertation focus on several important problems, such as mortality graduation, mortality prediction, insurance pricing and ruin probability, and proposes corresponding models and computational algorithms. Main contents of this dissertation are as follows:In Chapter 1, the background and motivation of this dissertation are introduced together with a review of some existing methods and models for survival analysis and risk theory. Finally, main research work of this dissertation is discussed.Chapter 2 is devoted to introducing the entropy concept and entropy optimization principles, which are the tools for predicting the probability distribution. Based upon the duality theory of convex programming, we derive the corresponding dual programs of two entropy optimization problems. Finally, we introduce two methods (the entropy regularization and the exponential penalty) for solving the min-max problem, and explore the duality relationship between these two approaches.In Chapter 3, the graduation and prediction models for mortality are studied. Based on two entropy optimization principles, we present two graduation models and one mortality predicting model that include the maximum entropy graduation model, the multi-objective graduation model and the minimum cross entropy prediction model, respectively. They improve the existing methods in computational precision and efficiency.In chapter 4, the concept of informational entropy is fully explored in the context of insurance premium principles. An additional term of premium is added to the mean-variance based pricing models, in order to cover the systematic risk arising in the selection of probably distributions.In chapter 5, we propose a new premium approach, the entropy regularization method, and derive an exponential-type premium formula. The main feature of this new approach lies in the fact that a probability transformation process, from anestimated one into a "real" one, is embedded in the premium derivation and transformation formula is explicitly given. This does not only make the present approach easily understood, but also establishes a firm link of it with some famous probability transformation and premium principles.In chapter 6, the large deviation theory is employed in risk theory. The ruin probability for the surplus progress is derived based upon the Cramer theorem. Two approaches to obtaining the rate function are discussed, of which the one is relied on Chernoff formula and the other is through the cross entropy. In particular, the latter method can be used to obtain the rate function under the complex probabilities.The last chapter gives a summary of the dissertation and some possible extensions of the present work.