Optimal Investment And Risk Management Policies For An Insurance Company With Ambiguity

It is of great important to assure the reasonable decisions for an insurance company to operate its business.In recent years,many researchers have studied the optimal decision problems under the assumption of that the decision makers know accurately of the probability measure used in the model setting.A few researchers have paid their attention to the optimal decision problems with ambiguity.However,these problems can't be ignored.It is more realistic with consideration of ambiguity than without it.In fact,many researchers have paid attention to the optimal control problems.They have studied optimal investment,optimal dividend,optimal premium control,optimal reinsurance and so on.Generally,the objectives of these studies are maximize expected utility,minimize the probability of ruin and maximize dividends,etc.Researchers often construct models by data and various technologies.Actually,the model that is estimated by some data is highly possible misspecification.If misspecification problem is ignored,the optimal strategy will be inaccurate or far from the reality.In general,there are two ways to amend potential misspecifications.One is that one can enhance the amount and quality of the data or improve the technology of simulation.Another one is that one can investigate the optimal control problems with consideration of the existence of ambiguity.In this paper,we study the optimal control problem with the objective of maximizing the expected utility and minimizing the probability of ruin with consideration of the existence of ambiguity.This paper will provide some suggestions for insurance companies.In this thesis,we consider three decision problems with ambiguity aversion,i.e.,investment,reinsurance and premium control problems.These are studied from two perspectives,i.e.,maximizing the expected utility of terminal wealth and minimizing the probability of ruin respectively.We investigate the optimal investment problem,optimal investment and premium control problem and optimal investment and reinsurance problem with ambiguity aversion under the perspective of maximizing the expected utility of terminal wealth.We study the optimal investment policies for insurers who are concerned about the model misspecification under risk control context.It should be noted that only a few researchers have investigated optimal robust control problem from the perspective of minimizing ruin probability.In addition,researches relating to these problems from insurance companies' standpoints are rarely seen.We mainly study the decision-making problems of insurance companies from the perspectives of investment returns and risk management.This thesis contains seven chapters.In Chapter 1,we give the introduction.In Chapter 2,we describe the basic knowledge and basic model.There are main research problems in Chapter 3,4,5 and 6.The investment policies and risk management policies are studied in these chapters.We give some conclusions and research plans in Chapter 7.Chapter 1.Introduction.We introduce the research background and significance,research ideas and methods,research features and innovations,and literature review.In this paper,we display the literature review from the perspectives of the research history of classical risk model,the investment policy and the risk management policy under the assumption of accurate model,the historical background and development of model ambiguity respectively.Chapter 2.Basic model and model ambiguity.The basic knowledge,the foundation models of the insurance company,and the price process of risky asset of the financial market are given in this chapter.We present the model ambiguity and the derivation of model ambiguity.Chapter 3.Optimal investment policy with ambiguity aversion.In this chapter,we investigate the optimal investment policy for an insurer with ambiguity aversion.The closed-form solutions of optimal investment policy and value function are obtained.Furthermore,we compare the results between us and Browne's(1995)Chapter 4.Optimal investment and premium policies with ambiguity aversion.Because of the complexity of financial market,the model of the financial market may prone to ambiguity.Hence,it is more realistic to investigate the optimal investment and premium control for insurers with consideration of the existence of ambiguity in the financial market.Intuitively,the insurance company will consider the ambiguity in the financial market,while the surplus process of the insurance company will be considered completely correct due to its long-time applying,operating and testing.Under the assumption of above,the closed-form expressions for the optimal investment policy,the optimal premium control policy and the value function are obtained with maximizing the expected utility of the terminal company's wealth and with ambiguity aversion.By the results,there is a relationship on the optimal investment policies between considering the ambiguity aversion and without considering the ambiguity for the financial market model.In particular,the results in this paper contain some findings of Zhou et al.(2017).Chapter 5.Optimal reinsurance and investment policies for an insurer with ambiguity aversion: variance premium principle.We consider the optimal reinsurance and investment problems for an insurer in three situations.Specially,instead of the expected premium principle,we assume the reinsurance premium is calculated via the variance premium principle.We firstly consider the optimal reinsurance problem when there is no risk asset in the market.And then,we investigated the optimal reinsurance and investment policies under the assumption of that there is an accurate modeled risky asset in the financial market.Furthermore,we study the optimal reinsurance and optimal investment problem for an insurer who worries about model ambiguity.By the dynamic optimal principle,we derive the Hamilton-Jacobi-Bellman(HJB)equation of value functions and give the closed-form solutions of optimal reinsurance and investment policies in three situations.Our main work is to investigate the impacts of the existence of ambiguity on optimal policies.In particular,we compare the optimal policies in three situations.Chapter 6.Optimal investment policies for insurers with ambiguity: minimizing the probability of ruin.We study the optimal investment policies for insurers who are concerned about the model misspecification under risk control context.It should be noted that only a few researchers have investigated optimal robust control problem from the perspective of minimizing ruin probability.In addition,researches relating to these problems from insurance companys' standpoints are rarely seen.By assuming that ambiguity only exists in the financial model and applying stochastic control method,we obtain the Hamilton-Jacobi-Bellman(HJB)equation associating to the value function under the objective of minimizing the probability of ruin.Most interesting,we also investigate the optimal investment policy that considers the ambiguity not only exists in the financial model but also exists in the insurance model.The closed-form solutions of optimal investment policies and value functions are obtained in both of the above situations.In analyzing the value function,we show that insurers should pay more attention to the insurance market and should not over-relied on the financial market.Chapter 7.Conclusions and research plans.We give some conclusions and research plans in this chapter.This thesis contains the following innovations.1.This paper studies the optimal decision-making problem under model ambiguity from the perspective of risk control.Many researchers considered the optimal control problems without considering the possibility of model ambiguity.Even a few researchers considered the optimal control problem with model ambiguity,they also aimed to maximize expected utility of wealth.According to my knowledge,no one has studied the optimal control problem for an insurer with ambiguity from the perspective of minimizing probability of ruin.An insurer should make decisions not only from the perspective of maximizing the expected utility of wealth,but also from the perspective of risk control(minimizing the probability of ruin).Therefore,it is necessary to study the optimal decision-making problem with model ambiguity from the perspective of risk control.This paper discusses the decision-making problems of insurers with consideration of model ambiguity from the perspective of minimizing ruin probability.This is an important innovation of this thesis.From the perspective of minimizing the probability of ruin,that is of great significance for insurance companies to control risk.2.The impacts of model ambiguity on insurers between in financial market and in the insurance market models are compared.In general,the researchers study the decision-making problems with consideration of the existence of ambiguity that only exists in the financial market.This paper considers the problems with ambiguity that not only exists in the financial market but also exists in the insurance market.Furthermore,we compare the impacts of the ambiguity between them.In comparing the impacts,we show that insurers should pay more attention to the insurance market and should not over-relied on the financial market.3.Optimal decision-making problems for an insurer with ambiguity aversion: variance premium principle.Basing on previous works,we note that most of the researchers also take the expected value principle to calculate reinsurance premiums.However,the variance premium principle is also a very popular way for premium calculations,especially for non-life insurance.The advantage of the variance premium principle is that it not only refers to the expectation of risk,but also refers to the variance of risk.So far,only a few studies adopted it for risk control in a dynamic setting.In our work,we take variance premium principle and consider the optimal reinsurance and optimal investment problems for an insurer in three cases progressively.By employing the dynamic programming principle,we give the analytic solutions of the value functions and optimal policies in all three cases.4.Optimal investment and premium control for insurers with ambiguity.In an insurance company,we can observe that the claim arrival rate depends heavily on the premium rate.Naturally,once the premium rate is changed,the claim arrival rate will change.As a result,there is a monotone function describing the relationship between the premium rate and the claim arrival rate.Furthermore,either the premium rate or the claim arrival rate which depicts the premium income will be served as a control variable.In this paper,we consider the optimal investment and premium control problem for insurers who worry about model ambiguity.Our purpose is to find the impacts of model ambiguity and the impacts of the correlative coefficient on the optimal policies.5.We obtain the closed-form solutions and give some numerical examples and economic explanations for the four optimal control problems.In this thesis,we investigate the optimal investment problem,optimal investment-premium problem,optimal investment-reinsurance problem and optimal investment problem with minimizing the probability of ruin.We obtain the closed-form solutions of optimal policies and value function.And we analyze the impact factors which can affect the above optimal control problems.The economic explanations are given to explain the results.