  Viewing stochastic processes through the lens of occupation measures has
proved to be a powerful angle of attack for the theoretical and computational
analysis of a wide range of stochastic optimal control problems. We present a
simple modification of the traditional occupation measure framework derived
from resolving the occupation measures locally on a partition of the control
problem's space-time domain. This notion of local occupation measures provides
fine-grained control over the construction of structured semidefinite
programming relaxations for a rich class of stochastic optimal control problems
with embedded diffusion and jump processes via the moment-sum-of-squares
hierarchy. As such, it bridges the gap between discretization-based
approximations to the solution of the Hamilton-Jacobi-Bellmann equations and
approaches based on convex optimization and the moment-sum-of-squares
hierarchy. We demonstrate with examples that this approach enables the
computation of high quality bounds on the optimal value for a large class of
stochastic optimal control problems with notable performance gains relative to
the traditional occupation measure framework.
