An impulse control of a geometric Brownian motion with quadratic costs

We examine an optimal impulse control problem of a stochastic system whose state follows a geometric Brownian motion. We suppose that, when an agent intervenes in the system, it requires costs consisting of a quadratic form of the system state. Besides the intervention costs, running costs are continuously incurred to the system, and they are also of a quadratic form. Our objective is to find an optimal impulse control of minimizing the expected total discounted sum of the intervention costs and running costs incurred over the infinite time horizon. In order to solve this problem, we formulate it as a stochastic impulse control problem, which is approached via quasi-variational inequalities (QVI). Under a suitable set of sufficient conditions on the given problem parameters, we prove the existence of an optimal impulse control such that, whenever the system state reaches a certain level, the agent intervenes in the system. Consequently it instantaneously reduces to another level.