| One of the significant topics in finance is the allocation of endowments among assets with the purpose of achieving higher returns during the investment horizon.On the basis of previous work,this thesis attempts to study the optimal portfolio problems of one investor and two investors respectively with the extended form of Merton’ utility function.First,for the case of one investor,we assume that the investor can invest in n risky assets which evolve according to the exponential Ornstein-Uhlenbeck process and a riskless asset.Inspired by Kraft and Ye L.L.,we select the extended form of the utility function constructed by Merton which attaches the risky assets to the discount rate.Under the above assumptions,we study the corresponding portfolio optimization problem.We formulate the problem applying the dynamic programming principle,deriving the Hamilton-Jacobi-Bellman(HJB)equation.With the assistance of the Girsanov theorem and the Feynman-Kac theorem,we obtain the explicit solution of the optimal investment strategy.Then,we extend the problem from one investor to two competing investors under the framework of the above price models and utility function.In this case,we only consider the two investors investing in one riskless asset and one risky asset.By drawing on the work of Espinosa and Touzi,we suppose the two investors pay attention to not only their terminal wealth but also the terminal relative performance measured by the difference between their terminal wealth.The prime objective of each investor is to maximize the discounted expected utility of a weighted average of his terminal wealth and the difference between their terminal wealth.The problem of finding optimal strategies for the both investors is modeled by a non-zero-sum stochastic differential investment game.We formulate the problem applying the dynamic programming principle,bringing about,the two coupled Hamilton-Jacobi-Bellman(HJB)equations.By utilizing the Girsanov theorem and the Feynman-Kac theorem,we deduce the explicit solutions for the investment strategies of two competing investors.Futhermore,for the sake of observing the impact of the sensitivity of each investor to the performance of his competitor on his equilibrium strategy,we provide a simulation result,together with sound economic interpretations. |
