Stochastic maximum principle with control-dependent terminal time and applications

In this thesis we study stochastic control problems with control-dependent stopping terminal time. We assess what are the methods and theorems from standard control optimization settings that can be applied to this framework and we introduce new statements where necessary. In the first part of the thesis we study a general optimal liquidation problem with a control-dependent stopping time which is the first time the stock holding becomes zero or a fixed terminal time, whichever comes first. We prove a stochastic maximum principle (SMP) which is markedly different in its Hamiltonian condition from that of the standard SMP with fixed terminal time. The new version of the SMP involves an innovative definition of the FBSDE associated to the problem and a new type of Hamiltonian. We present several examples in which the optimal solution satisfies the SMP in this thesis but fails the standard SMP in the literature. The generalised version of the SMP Theorem can also be applied to any problem in physics and engineering in which the terminal time of the optimization depends on the control, such as optimal planning problems. In the second part of thesis, we introduce an optimal liquidation problem with control-dependent stopping time as before. We analyze the case when an agent is trading on a market with two financial assets correlated with each other. The agent’s task is to liquidate via market orders an initial position of shares of one of the two financial assets, without having the possi- bility of trading the other stock. The main results of this part consist in proving a verification theorem and a comparison principle for the viscosity solution to the HJB equation and finding an approximation of the classical solution of the Hamilton-Jacobi-Bellman (HJB) equation associated to this problem.Open Acces