Backward Optimal Investment Under The Optimal Control Of Stochastic Differential Equations, Differential Game, And Entropy Risk Constraints

This thesis investigates the mixed optimal control and the zero-sum differential game on the switching and stopping of backward stochastic differential equations (BSDEs,for short),and the optimal investment with an entropic risk constraint.It consists of two parts.Part 1 is concerned with the mixed optimal control and the zero-sum differential game on the switching and stopping of BSDEs.It consists of Chapters 2,3 and 4. In Chapter 2,we model and solve the switching and stopping of a project as a real option.In Chapters 3 and 4,the problem discussed in Chapter 2 is generalized to the switching and stopping of general BSDEs whose generator and terminal value are allowed to be switched among a finite number of given modes with a positive cost being incurred for each switching.By distinguishing the coincidence and the contradiction between the switcher's and the stopper's purpose,we study the mixed optimal control and the zero-sum differential game therein.The optimal control is obtained for the former problem and the existence of the value process is proved for the latter.The respective HJB equation and HJBI equation of both problems turn out to be multi-dimensional reflected BSDEs with one-sided and two-sided hybrid barriers.For the multi-dimensional reflected BSDEs with one-sided hybrid barriers, we prove the existence of the solution by the penalization method,and obtain the uniqueness of the solution by linking the first component of the solution to the value process of the mixed optimal control problem.For the multi-dimensional reflected BSDEs with two-sided hybrid barriers,we prove the existence of the solution by the Picard iteration method,invoking the generalized monotonic limit theorem,and obtain the uniqueness of the solution by linking the first component of the solution to the value process of the optimal switching problem of one-dimensional reflected BSDEs.In Chapter 5,the existence of the solution to the multi-dimensional reflected BSDEs with two-sided hybrid barriers is further proved by a penalization method under some slight different assumptions. Part 2 is concerned with the optimal investment with a constraint on the entropic risk for the terminal wealth.It consists of Chapters 6 and 7 In Chapter 6,we introduce the concepts of coherent risk measure and convex risk measure, then generalize the definition of entropic risk measure and examine the properties of entropic risk measure.In Chapter 7,we study the optimal portfolio selection with a risk constraint which is expressed in terms of the entropie risk measure.By examining the dual problem,we prove the existence and uniqueness of the solution, and obtain an expression for the optimal terminal wealth profile.