This technical note develops a unified methodology for probabilistic analysis and optimal control design for jump diffusion processes defined by polynomials. The statistical moments of these systems can be described by a system of linear ordinary differential equations. Typically, however, the low-order moments depend on higher order moments, thus requiring an infinite system of equations to compute any moment exactly. Here, we develop a methodology for bounding statistical moments by using the higher order moments as inputs to an auxiliary convex optimal control problem with semidefinite constraints. For steady-state problems, the auxiliary optimal control problem reduces to a static semidefinite program. The method applies to both controlled and uncontrolled stochastic processes. For stochastic optimal control problems, the method gives bounds on achievable performance and can be used to compute approximately optimal solutions. For uncontrolled problems, both upper and lower bounds on desired moments can be computed. While the accuracy of most moment approximations cannot be quantitatively characterized, our method guarantees that the moment of interest is between the computed bounds.