Investigation Of Continuous-time Portfolio Selection Under Mean-variance Criterion

Mean-variance analysis for optimal asset allocation is one of the classical resultsof financial economics. It is well known that the mean-variance approach proposed byMarkowitz is viewed as the foundation of modern finance theory. After the originalpublication in Markowitz (1952), a vast number of papers have been published on thistopic. Most of these papers deal with the single period case. It is very easy to see thatthe optimal portfolio problem in a multi period framework and continuous time versionis time-inconsistent in the sense that the Bellman Optimality Principle does not hold.Nowadays there are two basic ways of handling time-inconsistency in optimalcontrol problems in the literature. The first method is to make the decision makersthemselves precommitted to follow the policies chosen at the initial time in the fu-ture. The second method is to take the time-inconsistency more seriously and study theproblem within a game theoretic framework. One possible interpretation of the time-inconsistency is that our preferences change in a temporally inconsistent way as timegoes by and we can thus view the mean-variance problem as a game, where the play-ers are the future incarnations of our own preferences. Nash equilibrium points can befound in the game theoretic approach to address the general time-inconsistency.Chapter2investigates the precommitted strategy and time-consistent strategy re-spectively for a dual risk model. We assume that the company can invest into a risk-freeasset and a risky asset. Short-selling and borrowing money are allowed. Due to lackof iterated-expectation property, the Bellman Optimization Principle does not hold. ALagrange method is proposed to derive the precommitted investment strategy. Basedon the same discussions in Bj o¨rk and Murgoci (2010), the time-consistent investmentstrategy is also obtained. By comparing the precommitted strategy with time-consistentstrategy, we find that the diferent strategies have diferent advantages: the former canmake value function maximized and the latter strategy is time-consistent.Chapter3constructs a general risk model with difusion by incorporating the clas-sic risk model and the dual risk model and investigates the precommitted strategy andthe time-consistent strategy for this general risk model under mean-variance criterion,respectively. We assume that the company can invest in a financial market which con- sists of a risk-free asset and multiple risky assets. We can derive the precommittedinvestment strategy by the Lagrange method stated in Chapter2. Meanwhile from thegame theoretical perspective, we find the time-consistent investment strategy by solv-ing the extended Hamilton-Jacobi-Bellman equations. By comparing the precommittedstrategy with the time-consistent strategy, we find that the company under the time-consistent strategy has to give up the better current utility in order to keep a consistentsatisfaction over the whole time horizon.Chapter4extends the portfolio management stated in Bjo¨rk, Murgpci, Zhou(2012)to a financial market which consists of a risk-free asset and multiple risky assets andinvestigates the optimal investment strategies including the precommitted strategy andtime-consistent strategy with a state dependent risk aversion coefcient. Furthermore,we theoretically and numerically provide the efect of the parameters on the two optimalstrategies and the corresponding value functions.Chapter5investigates the optimal investment strategies for a dual risk model and adual approximation difusion model under a financial market with jumps which impliesthe price of risky assets are subject to stochastic diferential equations with jumps, andobtains the time-consistent strategies from the game theoretic point of view.