  We develop a general approach to the Policy Improvement Algorithm (PIA) for
stochastic control problems for continuous-time processes. The main results
assume only that the controls lie in a compact metric space and give general
sufficient conditions for the PIA to be well-defined and converge in continuous
time (i.e. without time discretisation). It emerges that the natural context
for the PIA in continuous time is weak stochastic control. We give examples of
control problems demonstrating the need for the weak formulation as well as
diffusion-based classes of problems where the PIA in continuous time is
applicable.
