Study On Large Deviations For Heavy-Tailed Risk And Portfolio Decisions In Continuous Time

This paper consists of two parts:the large deviations of the web dependent and heavy-tailed risk series and the optimal investment consumption problem in the con-tinuous time market.With the development of economy,the financial and insurance markets present increasingly complex interdependencies,and the dependent structure of risk sequences becomes more complex.How to describe various types of dependent risks is an impor-tant issue that needs to be addressed in financial and insurance risk management,and is also a hot topic in academic research.In recent years,extreme financial and insur-ance phenomena have occurred frequently,causing the market to face great risks and causing huge losses for investors.Heavy-tailed distribution can characterize the loss characteristics of some extreme events,and large deviation theory is a powerful tool for portraying extreme risks.It is of great theoretical and practical significance to study the large deviation of heavy tail risk based on dependent structures.Chapter 2 of this thesis first constructs the web dependent structure based on the actual situation,and studies the web renewal counting process based on the web dependent structure.Chapter 3 gives the precise large deviation results of two class of heavy tail distributions for the web renewal risk model.Chapter 4,for the heavy-tailed risk of consistently-varying tailed distributions,studies the moderate deviations and the precise large deviations for the web renewal risk model.The optimal investment and consumption problem is a major content that directly affects the interests of investors,and is one of the core contents of financial market study.It is of great significance to study the problems of optimal investment and con-sumption,which provides investors with decision-making reference and also promotes the development of relevant theories.The second part of this thesis studies the optimal investment and consumption selection by expected utility maximization in the contin-uous time dynamic setting which serves as a fundamental problem in mathematical finance.The dual theory and martingale method are used to derive the optimal strate-gies.Assume that market is complete and the price of risk assets are characterized by diffusion processes.From the perspective of maximizing the expected terminal wealth utility,chapter 5 considers the model that the drift coefficient of the risk asset price to obey a general diffusion process,obtains the general solution to the portfolio choice problem.Using the confluent hypergeometric functions,the exact expressions of the optimal investment strategies are given for three special processes:O-U process,ge-ometric Brownian motion and a-hypergeometric process.Chapter 6 investigates the investment and consumption selection from the perspective of behavioral finance.For a loss aversion investor and the given S-shaped utility function,Chapter 6 obtains the optimal investment and consumption strategies by maximizing the expected consump-tion utility,and compares them with the classical results through numerical methods.