An impulse control of a geometric Brownian motion with quadratic costs Masamitsu Ohnishi a,b, * , Motoh Tsujimura b a Graduate School of Economics, Osaka University, 1-7 Machikaneyama-machi, Toyonaka, Osaka 560-0043, Japan b Daiwa Securities Chair, Graduate School of Economics, Kyoto University, Yoshida-hommachi, Sakyo-ku, Kyoto 606-8501, Japan Abstract 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 for- mulate 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 con- trol such that, whenever the system state reaches a certain level, the agent intervenes in the system. Consequently it instantaneously reduces to another level. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Impulse control; Quasi-variational inequalities; Quadratic costs 1. Introduction Transaction costs have been studied by many authors in mathematical finance and economics. See, for example, Cadenillas [4]. In fact we can observe the several transaction costs such that monitoring costs, trading costs and so on. In general there are two types of transaction costs. One is proportional transaction costs which are depend on the magnitude of control. The other is fixed transaction costs which are inde- pendent of the magnitude of control. When we examine optimization problems of agents under uncertainty, we can solve the problems by using stochastic control theory. See, for examples, Fleming and Rishel [9] and 0377-2217/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2004.07.006 * Corresponding author. E-mail addresses: ohnishi@econ.osaka-u.ac.jp (M. Ohnishi), tsujimura@econ.kyoto-u.ac.jp (M. Tsujimura). European Journal of Operational Research 168 (2006) 311–321 www.elsevier.com/locate/ejor