Study On Asymptotic Analysis On The Renewal Risk Model With Investment Returns And Its Application

Extreme events, including extreme natural risks and extreme financial risks, often lead to serious consequences. In the insurance industry, lots of significant insurance risks are just huge claims caused by a major disaster event. A catastrophe claims can lead to Insufficient solvency of the insurance company or even bankruptcy; As the coun-terpart, financial risks mainly include investment risks of insurance funds and interest rate fluctuation risks, and the consequence of an investment failure is catastrophic cor-respondingly. How to deal with and prevent extreme risk is one of the key problems in modern risk management. In the classical risk model, Investment factors of financial derivatives are often not considered. However, on the one hand, with the development of the insurance industry in the modern society, its total assets continue to expand and has the ability to invest huge sums of money in financial markets; on the other hand, owing to increasingly diverse investment channels by continuous innovation of financial derivatives, with abundant capital and professional investment department, insurance companies have become the most important institutional investor in the financial mar-ket. Therefore, it is time to take into account the measurement and management of financial risks brought by investment in capital markets besides considering traditional claim risk, because it directly related to the solvency of insurance companies. As a result, risk models with insurance risk and financial risk have emerged as the times require, and have gradually become an important branch of risk theories.Ruin probability is the core concept of risk theory, which is an important indicator to measure the stability of insurance companies. Probability distribution of extreme events is often described to be heavy-tailed. Precise large deviations of heavy-tailed distributions provides a theoretical basis for risk measurement associated with rein-surance. In this paper, we study the asymptotic estimation of ruin probabilities and precise large deviations of the present value of the claim under the framework of the heavy-tailed integrated risk model. In addition, we analyse various factors which have impacts on surplus processes in this model with simulation. Finally, we apply the the-oretical results to the risk measure and the optimal investment strategy. The details are as follows:In chapter 1, we state research background and research significance. we demon-strate the rationality and feasibility of the subject selection from the perspective of models and problems. After briefly describing the research context and research status of risk models, the main content of this dissertation are proposed.In chapter 2, we give the preliminary knowledge which is necessary for the main result of the dissertation. It mainly covers the following aspects. The first is relevant theory about Lévy process, containing Lévy-Khinchine representation theorem, Lévy-It(?) decomposition, classification of Lévy processes, It(?) formula of Lévy processes and stochastic exponential and so on. Secondly, we introduce the definition and classifica-tion of heavy-tailed distributions, especially including properties of the subexponential distribution class, the dominant variation class and the regular variation class.In chapter 3, we study the uniformly asymptotic estimation of the finite time ruin probability in the heavy-tailed integrated risk model with time dependent structure. Firstly, the integrated risk model is introduced, which assumes that a insurance compa-ny invests its reserve both in a bond and in a stock under a constant mix strategy. The risky asset is modelled by a exponential Lévy process and the bond brings a constant interest rate. Secondly, Specific dependence structure is imposed on claim sizes and corresponding inter-arrival times, and its rationality is verified under common copula structures. Then, assuming claim sizes belong to L∩D, uniformly asymptotic estima-tion of the finite time ruin probability are obtained under the pure risk investment case and the mixed case respectively. results under the both cases are compared from the perspective of conditions and conclusions. Moreover, application on risk managements are also introduced briefly. Finally, main results are proved by a series of lemmas.In chapter 4, precise large deviations of discounted values of claim sizes are mainly discussed. The return of asset portfolio is still modelled by a exponential Lévy process. Furthermore, independence structure is imposed on claim sizes and corresponding inter-arrival times. Then, assuming that claim sizes belong to R-α where α> 2 and 1<α<2 are considered respectively, precise large deviations of discounted values of claim sizes are obtained under the dangerous insurance risk case. Based on these results, we pointed out that the conclusion for the stochastic discount value of net claims is also established. According to the different range of regular variation index, the methods of proof we adopt are different due to the differences in the properties. Especially, in the case where expectation is finite but variance is infinite, the double truncation technique is utilized in addition to some properties of the regular variation class. Finally, main results are proved by a series of lemmas.In chapter 5, the surplus process of integrated risk models is analyzed by stochastic simulation firstly. By setting Different situations such as the investment processes, claim amount distributions and investment strategies, influences on the surplus process are qualitative analysed. secondly, according to the relevant conclusions, the optimal investment strategy integrated risk models in a given period of time is considered, where the two constraint conditions are imposed respectively, one is the variance of surplus in the end of this period, the other is the finite time ruin probability. Numerical analysis is implemented in the different scenarios which have impact on the optimal investment strategy. The results show that relative to the mean variance model, the constraint of finite time ruin probability is more suitable for risk management of the insurance company.In chapter 6, the research works and main innovative points are summarized, and then some problems need to be improved and studied are pointed out.