A Finite Horizon Optimal Stochastic Impulse Control Problem with A Decision Lag

TL;DR

This paper investigates a finite horizon stochastic impulse control problem with a fixed decision lag, establishing the value function's properties, deriving the HJB equation, and characterizing the optimal control as a viscosity solution.

Contribution

It introduces a novel impulse control model with decision lag, proves the value function's continuity, and characterizes it via a unique viscosity solution to the HJB equation.

Findings

Proved the continuity of the value function.

Derived the HJB equation with special features.

Constructed optimal impulse controls based on the value function.

Abstract

This paper studies an optimal stochastic impulse control problem in a finite horizon with a decision lag, by which we mean that after an impulse is made, a fixed number units of time has to be elapsed before the next impulse is allowed to be made. The continuity of the value function is proved. 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 exhibit some special feature of the problem. 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 given. Moreover, a limiting case with the waiting time approaching $0$ is discussed.