On the LP formulation in measure spaces of optimal control problems for jump-diffusions

In this short note we formulate a infinite-horizon stochastic optimal control problem for jump-diffusions of Ito–Levy type as a LP problem in a measure space, and prove that the optimal value functions of both problems coincide. The main tools are the dual formulation of the LP primal problem, which is strongly connected to the notion of sub-solution of the partial integro-differential equation of Hamilton–Jacobi–Bellman type associated with the optimal control problem, and the Krylov regularization method for viscosity solutions.