  This paper develops a unified methodology for probabilistic analysis and
optimal control design for jump diffusion processes defined by polynomials. For
such systems, the evolution of the moments of the state can be described via a
system of linear ordinary differential equations. Typically, however, the
moments are not closed and an infinite system of equations is required to
compute statistical moments exactly. Existing methods for stochastic analysis,
known as closure methods, focus on approximating this infinite system of
equations with a finite dimensional system. This work develops an alternative
approach in which the higher order terms, which are approximated in closure
methods, are viewed as inputs to a finite-dimensional linear control system.
Under this interpretation, upper and lower bounds of statistical moments can be
computed via convex linear optimal control problems with semidefinite
constraints. For analysis of steady-state distributions, this optimal control
problem reduces to a static semidefinite program. These same optimization
problems extend automatically to stochastic optimal control problems. For
minimization problems, the methodology leads to guaranteed lower bounds on the
true optimal value. Furthermore, we show how an approximate optimal control
strategy can be constructed from the solution of the semidefinite program. The
results are illustrated using numerous examples.
