Smooth Fit Principle for Impulse Control of Multidimensional Diffusion Processes

Value functions of impulse control problems are known to satisfy quasi-variational inequalities (QVIs) [A. Bensoussan and J.-L. Lions, Impulse Control and Quasivariational Inequalities, Heyden & Son, Philadelphia, 1984; translation of Contrôle Impulsionnel et Inéquations Quasi Variationnelles, Gauthier-Villars, Paris, 1982]. This paper proves the smooth-fit C¹ property of the value function for multidimensional controlled diffusions, using a viscosity solution approach. We show by examples how to exploit this regularity property to derive explicitly optimal policy and value functions.