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On Near-Optimal Mean-Field Stochastic Singular Controls: Necessary and Sufficient Conditions for Near-Optimality

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  • Mokhtar Hafayed

    (Biskra University)

  • Syed Abbas

    (Indian Institute of Technology Mandi)

Abstract

Near-optimization is as sensible and important as optimization for both theory and applications. This paper deals with necessary and sufficient conditions for near-optimal singular stochastic controls for nonlinear controlled stochastic differential equations of mean-field type, which is also called McKean–Vlasov-type equations. The proof of our main result is based on Ekeland’s variational principle and some estimates of the state and adjoint processes. It is shown that optimal singular control may fail to exist even in simple cases, while near-optimal singular controls always exist. This justifies the use of near-optimal stochastic controls, which exist under minimal hypotheses and are sufficient in most practical cases. Moreover, since there are many near-optimal singular controls, it is possible to select among them appropriate ones that are easier for analysis and implementation. Under an additional assumptions, we prove that the near-maximum condition on the Hamiltonian function is a sufficient condition for near-optimality. This paper extends the results obtained in (Zhou, X.Y.: SIAM J. Control Optim. 36(3), 929–947, 1998) to a class of singular stochastic control problems involving stochastic differential equations of mean-field type. An example is given to illustrate the theoretical results.

Suggested Citation

  • Mokhtar Hafayed & Syed Abbas, 2014. "On Near-Optimal Mean-Field Stochastic Singular Controls: Necessary and Sufficient Conditions for Near-Optimality," Journal of Optimization Theory and Applications, Springer, vol. 160(3), pages 778-808, March.
    • Handle: RePEc:spr:joptap:v:160:y:2014:i:3:d:10.1007_s10957-013-0361-1
    DOI: 10.1007/s10957-013-0361-1
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    References listed on IDEAS

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    1. Buckdahn, Rainer & Li, Juan & Peng, Shige, 2009. "Mean-field backward stochastic differential equations and related partial differential equations," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3133-3154, October.
    2. L. P. Pan & K. L. Teo, 1999. "Near-Optimal Controls of a Class of Volterra Integral Systems," Journal of Optimization Theory and Applications, Springer, vol. 101(2), pages 355-373, May.
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    Cited by:

    1. Kuang, Daipeng & Li, Jianli & Gao, Dongdong & Luo, Danfeng, 2024. "Stochastic near-optimal control for a system with Markovian switching and Lévy noise," Chaos, Solitons & Fractals, Elsevier, vol. 178(C).
    2. Liangquan Zhang & Qing Zhou, 2018. "Near-Optimal Control of Stochastic Recursive Systems Via Viscosity Solution," Journal of Optimization Theory and Applications, Springer, vol. 178(2), pages 363-382, August.
    3. Mokhtar Hafayed & Syed Abbas & Abdelmadjid Abba, 2015. "On Mean-Field Partial Information Maximum Principle of Optimal Control for Stochastic Systems with Lévy Processes," Journal of Optimization Theory and Applications, Springer, vol. 167(3), pages 1051-1069, December.
    4. Yan Wang & Lei Wang & Kok Lay Teo, 2018. "Necessary and Sufficient Optimality Conditions for Regular–Singular Stochastic Differential Games with Asymmetric Information," Journal of Optimization Theory and Applications, Springer, vol. 179(2), pages 501-532, November.

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