Large Deviations, Risk Theory And Their Applications In Finance And Insurance

It is well known that the precise large deviations and the risk theory are two of main objects in insurance mathematics. We address in this thesis some important topics in insurance and finance, which are closely related to extremal value theory. They are precise large deviations for the aggregate claim amount process with heavy-tailed distributions, the Cramer-Lundberg approximation property and the Lundberg inequality for the ruin functions under the corresponding risk models.On one hand, some usual phenomenon in life to people is that there are some events hardly happen. However, the results of these events are not negligible. These kinds of events are the so-called extremal events in applied probability. For example, the extremal events in finance and insurance are those that take place with little probability, but will violently affect the whole insurance and finance system (or company) once they occur. Sometimes, the effects are destructive, such as hurricane, heavy traffic accident, fire and earthquake. These extremal events often ask for large claims. For an insurance company, statistically, the number of these events is only 20% to all of the claims number, but the claim size they needed comes to 80% in all the reparative money of the company. Inspired by it, the class of heavy-tailed distributions was introduced, which, in turn, greatly expanded modern large deviations and furthermore spurred the development of the renewal process, dynamical system, stochastic analysis, statistical mechanics, and financial mathematics (especially insurance mathematics) etc. Heavy-tailed distributions have found their wide applications in economic and social fields, such as finance stocks, risk investment, and statistical analysis.At present, the internal-and-external of heavy-tailed distributions (including the so-called regular variation, extended regular variation, sub-exponential distribution etc) and the large deviations of their partial sums (including precise large deviations, large deviation principle, moderate deviation principle etc) have been investigated widely and deeply under the case that summands are i.i.d. random variables.As an important part of the applied probability theory, large deviation principle is extremely useful in quantitatively describing extremal events. The formulation of the classical large deviation principles contributes to Cramer et al. But the random variables they concerned with had the light-tailed distributions (i.e., the moment functions of the random variables are finite). However, the heavy-tailed distributions are of great importance in the fields of finance and insurance, and many problems of them come down to one of large deviations (e.g. the problem of reinsurance). Therefore, the large deviations of partial sums and random sums of heavy-tailed random variables have become rational objects to the applied probability researchers.Nagaev, A.V. (1969a,b), Heyde (1967 a, b, 1968) and Nagaev, S. V. (1973, 1979) firstly established the precise large deviations for the partial sums of the independent random variables with a common heavy-tailed distribution. Cline & Hsing (1991) firstly proved the precise large deviations for the random sums of the independent randomvariables with a common heavy-tailed distribution. Kluppelbery et al. (1997) studied further and found the examples and -applications in the finance and insurance. Su et al. (2001) improved to assumptions in the results of Kluppelbery et al. (1997) and derived the precise large deviations for the renewal risk model with heavy-tailed claim sides.On the other hand, the risk theory, as an important theory in actuarial science, has become a popular subject of the research field. Simultaneously, as a vital method of measuring the risk of insurance, i.e., ruin theory, the ruin probability becomes a main object in the theory of risk. As we know, insurance is an important part of finance system and connects closely with the development of national economy and social insurance. However, It is a trade with the character o...