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

TL;DR

This paper formulates an infinite-horizon stochastic optimal control problem for jump-diffusions as a linear programming problem in measure spaces, demonstrating the equivalence of optimal value functions using duality and regularization techniques.

Contribution

It introduces a novel LP formulation for jump-diffusion control problems in measure spaces and establishes the equivalence of value functions via duality and viscosity solution methods.

Findings

Optimal value functions coincide for the LP and control problems.

Dual formulation relates to sub-solutions of Hamilton-Jacobi-Bellman equations.

Krylov regularization aids in analyzing viscosity solutions.

Abstract

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.