Optimal Investment Problems With Competition And Random Time Solvency Regulations

In this thesis,two optimal investment problems with competition and random time solvency regulations are studied.The first problem is that the investor aims at beating a benchmark policy within the regulation time.This problem is formulated as an active portfolio problem.By dynamic programming principle,the HJB equation associated with the control problem is derived,a verification theorem is also presented.Explicit solutions for optimal strategies are derived when the regulation time follows exponential distribution.Numerical schemes are provided when the regulation time follows more general distribution.The second problem is that the investor aims at beating an opponent who also dynamically adjusts his/her policies.This problem is formulated as a stochastic investment game problem within random time and regime switching.The HJBI equation associated with the game problem is derived,as well as a verification theorem.By solving an auxiliary problem,the explicit solutions for optimal policies of both investors when the regulation time follows exponential distribution.A pair of candidate optimal polices for both investors is conjectured as follows:given current external environment,one can use the parameters of the current environment and state of the system as decision variables to construct a single environment investment strategies.When the environment changes,the optimal strategies changes accordingly.It turns out that this kind dynamically changed single environmental strategies are optimal.