  We consider impulse control problems in finite horizon for diffusions with
decision lag and execution delay. The new feature is that our general framework
deals with the important case when several consecutive orders may be decided
before the effective execution of the first one. This is motivated by financial
applications in the trading of illiquid assets such as hedge funds. We show
that the value functions for such control problems satisfy a suitable version
of dynamic programming principle in finite dimension, which takes into account
the past dependence of state process through the pending orders. The
corresponding Bellman partial differential equations (PDE) system is derived,
and exhibit some peculiarities on the coupled equations, domains and boundary
conditions. We prove a unique characterization of the value functions to this
nonstandard PDE system by means of viscosity solutions. We then provide an
algorithm to find the value functions and the optimal control. This easily
implementable algorithm involves backward and forward iterations on the domains
and the value functions, which appear in turn as original arguments in the
proofs for the boundary conditions and uniqueness results.
