Research On Optimal Investment Portfolio And Optimal Insurance Strategy Problems With Stochastic Economic Factors

The problem of optimal investment occupies an important position in financial theory and practice.Whether it is for a normal investor or an insurer,selecting the optimal portfolio can always maximize the returns and minimize the risks of the investment.Take the insurer as an example,his/her investment with charging premiums could increase the profits and enhance the solvency.Therefore,it will be vital to choose the optimal investment strategy,while the investment operation of the insurer also plays an increasingly significant role.Another way to control risks is reinsurance.The insurer transfers part of the claimable loss(especially a substantial one)to the reinsurer through reinsurance,in order to lower the risks and stabilize the operations,making reinsurance an important means of risk control.In recent decades,with the continual developments of economy and finance,the risk of default has become increasingly prominent,which refers to the possibility of the borrower being unable to timely and fully repay the debt due to various reasons.Under the circumstance of a defaults,the investor would bear the economic loss caused by the inability to obtain expected returns,therefore it is also necessary for investors to take defaults into the account of optimal portfolio.Since the macroeconomic variations affect the asset prices then influence the investors’ risks,the consideration of stochastic economic factors(representing any macroeconomic variables leading to risk fluctuations)would be more reasonable to reflect the variations in the economic and financial markets.In addition,the mean-variance criterion is a very important standard for measuring risk,which is meaningful for insurers to measure risk,and also plays an important role in modern financial theory.Based on the above reasons,under the stochastic economic factors,we study the optimal investment and risk control in a Levy market,optimal investment and reinsurance,optimal investment and consumption strategies in a defaultable market,and optimal portfolio problem under the mean-variance criterion.Specifically,the main contents of this thesis are as follows.(1)For the problem of optimal investment and risk control for an insurer subject to a stochastic economic factor in a Levy market,a riskless bond and a risky asset are assumed to rely on a stochastic economic factor which is described by a Lévy SDE.The risk process is described by a "jump diffusion" SDE depending on the stochastic economic factor and is negatively correlated with capital gains in the financial market.According to the expected utility maximization criterion,we obtain the optimal strategies of investment and risk control under the logarithmic utility function and the power utility function,respectively.With the logarithmic utility assumption,we strengthen the budget constraint and the net profit condition and use the classical optimization method to obtain the optimal strategy.However,for the power utility function,we apply dynamic programming principle to derive the HJB equation,and analyze its solution in order to obtain the optimal strategy.We also show the verification theorem.Finally,numerical examples are presented to illustrate the impact of the market parameters on the optimal strategies.(2)For optimal investment and reinsurance strategies for an insurer with the stochastic economic factor,a risk-free asset and a risky asset are assumed to rely on a stochastic economic factor which is described by a diffusion process.We extend the classical risk theory,that is,the claim process considered is a compound Poisson process with the stochastic economic factor,and then use a Poisson random measure to express the compound Poisson process as a Levy process.By applying the expected utility maximization criterion,HJB equation under power utility function is derived by using dynamic programming principle.Then,by means of power transformation and the Feynman-Kac formula,we analyze and prove that the HJB equation has a unique classical solution,thus obtaining the optimal investment-reinsurance strategy,which is given in the verification theorem.Finally,sensitivity analysis is given to show the economic behavior of the optimal investment and reinsurance strategies.(3)For the issue of optimal investment and consumption strategies for an investor with the stochastic economic factor in a defaultable market,the price process is composed of a money market account and a default-free risky asset(such as a stock),assuming they rely on a stochastic economic factor described by a diffusion process.We also assume that the investor can invest in a defaultable perpetual bond,which is a jump process and is characterized by the reduced-form model.Both the default risk premium and the default intensity of the defaultable perpetual bond rely on the stochastic economic factor.The investor can also set a consumption ratio for consumption.Our goal is to maximize the infinite horizon expected discounted power utility of the consumption.Applying the dynamic programming principle,we derive the HJB equations and analyze them using the so-called sub-super solution method to prove the existence and uniqueness of their classical solutions.Next,we use a verification theorem to derive the explicit formula for optimal investment and consumption strategies.Finally,we analyze the sensitivity to parameters of the optimal control strategies and the value function through numerical simulation.(4)For optimal investment and risk control for an insurer under the meanvariance criterion with stochastic economic factor,in fact,we assume that the stochastic economic factor is characterized by jump-extended CIR(Cox-IngersollRoss)model.The financial market consists of a riskless bond and a risky asset,and the latter is related to the stochastic economic factor.We use a Levy SDE to describe the risk process we have,in which we extend the classic Cramér-Lundberg model to the Levy process,and additionally introduce the stochastic economic factor into this model.We assume that the insurer in question is a mean-variance optimizer.In other words,the decision that this insurer faces is to simultaneously maximize and minimize the mean and variance of his/her terminal wealth by selecting an optimal portfolio.We have uncovered closed-form solutions to mean-variance problems with respect to the efficient strategy and efficient frontier by solving for expected utility maximization of a quadratic function through the martingale method.Finally,we give a numerical example that analyzes the economic behavior of the efficient frontier.