Stochastic Near-Optimal Singular Controls for Jump Diffusions: Necessary and Sufficient Conditions

Abstract

This paper studies the necessary and sufficient conditions for near-optimal singular stochastic controls for the systems driven by non-linear stochastic differential equations with jump processes. The proof of our result is based on Ekeland's variational principle and some delicate estimates of the state and adjoint processes. We apply convex perturbation for continuous and singular components of the control. It is shown that optimal singular controls may fail to exist even in simple cases. This justifies the use of near-optimal stochastic singular controls, which exist under minimal hypothesis and are sufficient in most practical cases. Moreover, since there are many near-optimal singular controls, it is possible to choose suitable ones that are convenient for implementation. The set of controls under consideration is necessarily convex. We prove that under an additional hypothesis, the near-maximum condition on the Hamiltonian function is a sufficient condition for near optimality.