Singular control and impulse control: A common approach

Alain Bensoussan; John Liu; Jiguang Yuan

doi:10.3934/dcdsb.2010.13.27

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2010, Volume 13, Issue 1: 27-57. Doi: 10.3934/dcdsb.2010.13.27

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Singular control and impulse control: A common approach

1.
International Center for Decision and Risk Analysis, School of Management, University of Texas - Dallas
2.
Faculty of Business, The Hong Kong Polytechnic University, Hong Kong

Received: May 31, 2009

Revised: August 31, 2009

Published: June 2010

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Abstract

In this study, we investigate the connection between impulse control and singular control. The dynamic systems are driven by Brownian motion with drift. For simplicity we consider only one-dimension problems, where we can perform explicit calculations. We will see that Quasi-Variational Inequalities (QVI) are the common tool to consider these problems. The two problems have interesting links. By some aspects, singular control problems appear as particular cases of impulse control problems; however an impulse control is a particular case of singular control. We can, in particular, approximate an optimal singular control by a minimizing sequence of impulse controls. We show that optimal singular controls are linked to reflected diffusions. Thanks to the one-dimensionality we completely solve the QVI by the two band approach.

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Mathematics Subject Classification: Primary: 93E20, 60J70; Secondary: 90B05.

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