Research On Optimal Investment And Reinsurance For Complicated Jump-diffusion Risk Models Under Mean-Variance Criterion

Risk control and management has long been an important topic for insurance and financial firms.On the one hand,since the insurance and financial institutions have the opportunity to invest in the financial market,the problem of optimal investment has received great interest to them.On the other hand,insurers usually purchase reinsur-ance to transfer some of their own risks to another party at the expense of making less potential profit.Hence,finding optimal investment and reinsurance strategy to bal-ance the potential risk and profit has gained much attention in financial and actuarial literature.Compared with the utility maximization problems,the mean-variance criterion enables an insurer or investor to seek the lowest risk quantified by the variance of the return after specifying his/her acceptable return.Explicitly,this criterion not only captures the benefits but also considers the risks.Thus,for its rationality and practi-cability,the mean-variance criterion has become a rather popular criterion to measure the risk in finance theory and inspired numerous extensions and applications.It is well known that the mean-variance criterion lacks of iterated-expectation property,which gives rise to a time-inconsistent problem in the sense that the Bellman opti-mality principle is not feasible any more.There are two basic ways to handle the time inconsistency in optimal control problems.One is to study the pre-committed problem;another way is to take the time inconsistency more seriously and study the problem within a game theoretic framework.In this dissertation,several optimal investment and reinsurance problems under the mean-variance criterion are studied.By employing the techniques of the martin-gale approach,the Hamilton-Jacobi-Bellman(HJB)equation and the game theoretic framework,the optimal strategy and value function are derived and some numerical examples are presented to illustrate our results.The main work of this thesis includes the following parts:1.In Chapter 3,a problem of portfolio selection for a jump-diffusion model with constraint on the wealth process under the partial information is studied.As the wealth process is required to be no less than a given function,we tackle this problem via the technique of martingale approach by decomposing it into two steps:the first one is to obtain the optimal auxiliary terminal wealth,and the second is to find under what conditions this optimal auxiliary terminal wealth is attainable and then the optimal portfolio strategy is derived.By the tool of the filtering theory,we reduce the orig-inal partial information problem to a full information one.Closed-form expressions of the optimal strategy and the efficient frontier for both of the partial information and full information problems are obtained.Furthermore,we also discuss the case under no-shorting constraint and obtain the corresponding optimal results under the full information by transforming the problem into an equivalent one with constraint only on the wealth.However,this technique is not applicable any more for the partial information.To the best of our knowledge,this is the first work to consider the hidden Markov chain jump-diffusion model with the constraint on wealth process.2.In Chapter 4,the problem of optimal portfolio selection under the mean-variance utility are studied.It assumes that the financial market consists of one risk-free asset and two risky assets,whose price processes are given by jump-diffusion model and the two jump number processes are correlated through a common shock,and the Brownian motions are supposed to be dependent as well.Moreover,it is assumed that not only the risk aversion coefficient but also the market parameters such as the appreciation and volatility rates as well as the jump amplitude depend on a Markov chain with finite states.In addition,short selling is supposed to be prohibited.Because the mean-variance criterion lacks of iterated-expectation property,we solve the extended HJB equation within the game theoretic framework,and by the technique of stochastic control theory,we not only prove the existence and uniqueness of the solutions to a system of differential equations but also derive the explicit expressions of the optimal results.Some numerical examples are presented to illustrate the effects of parameters on the optimal strategy as well as the economic meaning behind.In the end,we discuss a general case of n(>3)risky assets and obtain a similar results if the Hessian matrix is a positive definite one.3.Chapter 5 is devoted to seek an optimal reinsurance strategy under a thinning-dependence risk model,in which the stochastic sources related to claim occurrence are classified into different groups,and that each group may cause a claim in each insurance class with a certain probability.Such a risk model is universal in reality.A typical example is that a severe car accident may cause not only the loss of the dam-aged car but also the medical expenses of injured driver and passengers.Under this criterion,different from the existing literature,we require that the reinsurance propor-tion belongs to[0,1],which makes this problem more challenging and complicated.Based on the technique of stochastic control theory and the corresponding extended HJB equation,explicit expressions of the optimal reinsurance strategy and the value function are derived for n=2 and some numerical examples are presented to illus-trate the effects of parameters on the optimal strategy.When n>3,we provide the methods of truncation and dimension reduction to get the optimal results.Specifical-ly,we give an example to show how the optimal results can be derived for n=3.As far as we know,this is the first work to consider the optimal reinsurance in the thinning-dependence structure by the technique of game theoretic framework.4.Instead of focusing on the pure financial or insurance market,Chapter 6 in-vestigates an optimal investment and reinsurance problem in a financial and insurance market,where the insurance risk model is modeled by a compound Poisson process and the price process of the stock is described by a jump-diffusion model.It is as-sumed that the aggregate claims process and the stock price process are correlated through a common shock,the risk aversion coefficient as well as the parameters of the risky asset are governed by a Markov chain,and short selling is not allowed as well.One of the main contributions in this chapter is that the effect of time-delay is taken into account.That is,the investment decision depends on not only curren-t market state but also previous information.Under the criterion of maximizing the mean-variance utility and within a game theoretic framework,explicit expressions of the optimal strategy and value function are derived.5.Furthermore,the problem of optimal time-consistent reinsurance and invest-ment in a jump-diffusion financial market with multiple assets is considered in Chapter 7.Due to the market without cash,the approach of separating the variables is out of work,by which the ODE system derived in the present chapter is highly nonlinear and we cannot guarantee the existence and uniqueness of a solution.Instead of ana-lyzing the ODE system,we utilize an alternative approach to solve the extended HJB equation.Closed-form expressions of the optimal strategy and the value function are derived,and the existence and uniqueness of the optimal results is proved as well.To illustrate our results more clearly,some special cases of our model and results are presented,and several numerical examples are provided to show the impacts of pa-rameters on the optimal strategy.Different from those existing literature,we obtain some new and interesting results.