Study Of Portfolio And Pricing Problems With Jump-Diffusion Processes

The portfolio selection and contingent claim pricing are always elementary problems in mathematical finance. almost conclusions of portfolio and contingent claim pricing base on the hypothesis that the price of risky securities follow diffusion processes. However, in reality market the price of securities is impacted by any important information, As a result, the price vary with discontinuous jump. It is not only simulate the reality market but also remedy insufficient in theory to adapt jump-diffusion processes. This dissertation studies portfolio selection and contingent claim pricing as the price of risky securities follow jump-diffusion processes. The dissertation is organized as follows. In chapterl, we discuss the origin of the study, the advance of the portfolio selection the origin of the study, the advance of the portfolio selection and contingent claim pricing. In chap-ter2,3.We study portfolio selection strategy to base on the safety first criteria. In chapter4, we study stochastic Differential portfolio Games for the prices of stocks with jump-diffusion processes, In chapter5, we study a contingent claim pricing and hedging in an incomplete market. In chaper6, we use the special change to study an Asia option pricing and hedging strategy. In chapter7. We study the pricing of the preferred hedging contingent claims under Transaction costs. The results are obtained in this dissertation include the following six aspects.(l)As the price of risky securities follow jump diffusion processes, we base the safety first model on portfolio selection first. Using stochastic filtering techniques and adapting martingale clarity method in full and partially observed complete and incomplete market the optimal portfolio sorcery is obtained. Final as the price o risky securities follow diffusion processes, the optimal strategy is obtained by stochastic analysis. The safety first model of portfolio selection is used to portfolio selection isn discrete time models, we extend the safety first criteria of Roy to the continuous-time security portfolio selection, our research remedy insufficient of Roy and Duan Li.(2)The thirty part research is continue to the second part research in the second research we give an hypothesis in function u(x)=-ωx2+λx , we assume ,if the hypothesis is not satisfied. the martingale method is not used to solve the problem of the second part as the state variable follow jump-diffusion processes, we set up the stochastic optimal control strategy, we adopt stochastic Lagrange method, the problem of the second part is special result of our conclusion.(3)There are many investors in reality market, How they keep beneficial status to adapt the portfolio selection strategy, we give an attempt research about the problem. The wealth's ratio of two investors is considered as the state variable as the logarithm utility function and general utility function are adapted to theproblem, the optimal strategy is obtained by stochastic optimal control method.(4)As the price of underlying securities follow jump-diffusion processes, we study pricing and hedging of a contingent claim the criteria of risk measure is presented by stochastic games, then optimal replication of contingent claim is obtained by the criteria. final, the optimal hedging strategy of a contingent claim is given by stochastic games, the pricing of a contingent claim is presented by the optimal wealth increase process in a stochastic volatility model.(5)As the price of underlying securities follow jump-diffusion processes the Asia option pricing is changed into a European option pricing by numeraire transformations and replication in complete market the Asia option pricing equation is obtained with an application of Merton method to hedge market risk. We give Asia option pricing to adapt a minimal equivalent martingale measure in a stochastic volatility model. As the price of underlying securities follow fractional Brownian motion the Asia option pricing equation and hedging strategy are obtained by fractional Ito theory and numeraire transformation...