Necessary Conditions for Optimal Control of Stochastic Systems with Random Jumps
SIAM Journal on Control and Optimization
Deterministic near-optimal control, part I: necessary and sufficient conditions for near-optimality
Journal of Optimization Theory and Applications
Singular Optimal Stochastic Controls II: Dynamic Programming
SIAM Journal on Control and Optimization
Deterministic near-optimal controls. Part II: dynamic programming and viscosity solution approach
Mathematics of Operations Research
Stochastic Near-Optimal Controls: Necessary and Sufficient Conditions for Near-Optimality
SIAM Journal on Control and Optimization
Near-optimal controls of a class of Volterra integral systems
Journal of Optimization Theory and Applications
Singular Stochastic Control, Linear Diffusions, and Optimal Stopping: A Class of Solvable Problems
SIAM Journal on Control and Optimization
Maximum Principle for Singular Stochastic Control Problems
SIAM Journal on Control and Optimization
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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.