Optimal Stochastic Impulse Control with Decision Lag

This dissertation studies the optimal stochastic impulse control problems with a decision lag, by which we mean that after an impulse is planned, a fixed number units of time has to be elapsed before the next impulse is allowed to be exercised. We present a series of results on the problems both in finite and infinite horizons. Also, some related results of mixed control policies are included. In more details, the continuity of the value function is proved first. Then a suitable version of dynamic programming principle is established, which takes into account the dependence of state process on the elapsed time. The corresponding Hamilton-Jacobi-Bellman (HJB) equation is derived, which exhibits some special feature of the problem. Further, the value function of this optimal impulse control problem is characterized as the unique viscosity solution to the corresponding HJB equation. An optimal impulse control is constructed provided the value function is determined. Moreover, a limiting case with the waiting time approaching 0 is discussed