Stochastic Maximum Principle for Optimal Liquidation with Control-dependent Terminal Time

In this paper 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. We present a simple example in which the optimal solution satisfies the SMP in this paper but fails the standard SMP in the literature.