Xin Guo
Multi-dimensional impulse control problems and their regularity
properties
Many problems in mathematical economics and finance are
formulated in the impulse/singular control framework. Compared to
regular controls, impulse control provides a more natural mathematical
framework when the state space is discontinuous. However, many
structural results amount to solving complex algebraic equations that
are hard to verify without a priori knowledge of the regularity
property, thus the correctness of the ``solutions'' is dubious. In this
talk, we provide sufficient conditions for the smooth-fit $C^1$
property of the value function for multi-dimensional controlled
diffusions, using a viscosity solution approach. This approach is
different from the Quasi-Variational Inequalities (QVI) established by
Bensoussan and Lions (1982). We show by simple examples where the
regularity property may fail especially in multi-dimensional case.