A Direct Solution Method for Stochastic Impulse Control Problems of One-dimensional Diffusions

Abstract

We consider stochastic impulse control problems where the process is driven by one-dimensional diffusions. Impulse control problems are widely applied to financial engineering and decision-making problems including the dividend payout problem, portfolio optimization with transaction costs, and inventory control. We shall show a new mathematical characterization of the value function in the continuation region as a linear function in a certain transformed space. The merits of our approach are as follows: (1) One does not have to guess optimal strategies or verify the optimality via a verification lemma, (2) the method of finding the solution (based on the new characterization of the value function) is simple and direct, and thereby (3) one can handle a broader class of reward and cost functions than the conventional methods that use quasi-variational inequalities.

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