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

Journal of Optimization Theory and Applications

Volumn 160, Issue 3, 2014, Pages 778-808

  Hafayed, Mokhtar ^a   Abbas, Syed ^b  

^a UNIVERSITY OF BISKRA   (Algeria)

^b INDIAN INSTITUTE OF TECHNOLOGY MANDI   (India)

Author keywords

Ekeland's variational principle. Generalized gradient; Mean field stochastic differential equations; Near optimal singular stochastic control; Necessary and sufficient conditions of near optimality

Indexed keywords

NONLINEAR EQUATIONS; STOCHASTIC CONTROL SYSTEMS; STOCHASTIC SYSTEMS; VARIATIONAL TECHNIQUES; VLASOV EQUATION;

CONDITION; EKELAND VARIATIONAL PRINCIPLE; EKELAND VARIATIONAL PRINCIPLE.; GENERALIZED GRADIENTS; MEAN-FIELD; MEAN-FIELD STOCHASTIC DIFFERENTIAL EQUATION; NEAR OPTIMALITY; NEAR-OPTIMAL; NEAR-OPTIMAL SINGULAR STOCHASTIC CONTROL; NECESSARY AND SUFFICIENT CONDITION OF NEAR-OPTIMALITY; SINGULAR STOCHASTIC CONTROL; STOCHASTIC DIFFERENTIAL EQUATIONS;

HAMILTONIANS;

EID: 84896492360     PISSN: 00223239     EISSN: 15732878     Source Type: Journal    
DOI: 10.1007/s10957-013-0361-1     Document Type: Article

References (37)

1. Pontryagin, L. S., Boltanskii, V. G., Gamkrelidze, R. V.: The Mathematical Theory of Optimal Processes. Interscience, New York (1962).
2. Bellman, R.: Dynamic Programming. Princeton Univ. Press, Princeton (1957).
3. Peng, S.: A general stochastic maximum principle for optimal control problems. SIAM J. Control Optim. 28(4), 966-979 (1990).
4. Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 353-424 (1974).
5. Gabasov, R., Kirillova, M., Mordukhovich, B. S.: The ε-maximum principle for sub-optimal controls. Sov. Math. Dokl. 27, 95-99 (1983).
6. Zhou, X. Y.: Deterministic near-optimal controls. Part I: necessary and sufficient conditions for near optimality. J. Optim. Theory Appl. 85, 473-488 (1995).
7. Zhou, X. Y.: Deterministic near-optimal controls. Part II: dynamic programming and viscosity solution approach. Math. Oper. Res. 21, 655-674 (1996).
8. Zhou, X. Y.: Stochastic near-optimal controls: necessary and sufficient conditions for near-optimality. SIAM J. Control Optim. 36(3), 929-947 (1998).
9. Hafayed, M., Abbas, S., Veverka, P.: On necessary and sufficient conditions for near-optimal singular stochastic controls. Optim. Lett. 7(5), 949-966 (2013).
10. Hafayed, M., Veverka, P., Abbas, S.: On maximum principle of near-optimality for diffusions with jumps, with application to consumption-investment problem. Differ. Equ. Dyn. Syst. 20(2), 111-125 (2012).
11. Chighoub, F., Mezerdi, B.: Near optimality conditions in stochastic control of jump diffusion processes. Syst. Control Lett. 60, 907-916 (2011).
12. Pan, L. P., Teo, K. L.: Near-optimal controls of class of Volterra integral systems. J. Optim. Theory Appl. 101(2), 355-373 (1999).
13. Huang, J., Li, X., Wang, G.: Near-optimal control problems for linear forward-backward stochastic systems. Automatica 46(2), 397-404 (2010).
14. Hui, E., Huang, J., Li, X., Wang, G.: Near-optimal control for stochastic recursive problems. Syst. Control Lett. 60, 161-168 (2011).
15. Zhou, X. Y., Sethi, S.: A sufficient condition for near-optimal stochastic controls and its application to manufacturing systems. Appl. Math. Optim. 29(1), 67-92 (1994).
16. Alvarez, L. H. R.: A class of solvable singular stochastic control problems. Stoch. Stoch. Rep. 67, 83-122 (1999).
17. Alvarez, L. H. R.: Singular stochastic control linear diffusion and optimal stopping: a class of solvable problems. SIAM J. Control Optim. 39, 1697-1710 (2001).
18. Cadenillas, A., Haussmann, U.: The stochastic maximum principle for singular control problem. Stoch. Stoch. Rep. 49(3-4), 211-237 (1994).
19. Bahlali, S., Chala, A.: The stochastic maximum principle in optimal control of singular diffusions with nonlinear coefficients. Random Oper. Stoch. Equ. 13, 1-10 (2005).
20. Bahlali, S., Mezerdi, B.: A general stochastic maximum principle for singular control problems. Electron. J. Probab. 10, 988-1004 (2005).
21. Dufour, F., Miller, B.: Maximum principle for singular stochastic control problem. SIAM J. Control Optim. 45(2), 668-698 (2006).
22. Dufour, F., Miller, B.: Singular stochastic control problem. SIAM J. Control Optim. 43(2), 708-730 (2004).
23. Karatzas, I., Ocone, D., Wang, H., Zervos, M.: Finite-fuel singular control with discretionary stopping. Stoch. Stoch. Rep. 71, 1-50 (2000).
24. Haussmann, U. G., Suo, W.: Singular optimal stochastic controls I, II. SIAM J. Control Optim. 33(3), 916-936, 937-959 (1995).
25. Kac, M.: Foundations of kinetic theory. In: Proc. 3rd Berkeley Symp. Math. Statist. Prob., vol. 3, pp. 171-197 (1956).
26. McKean, H. P.: A class of Markov processes associated with nonlinear parabolic equations. Proc. Natl. Acad. Sci. USA 56, 1907-1911 (1966).
27. Buckdahn, R., Li, J., Peng, S.: Mean-field backward stochastic differential equations and related partial differential equations. Stoch. Process. Appl. 119, 3133-3154 (2009).
28. Ahmed, N. U.: Nonlinear diffusion governed by McKean-Vlasov equation on Hilbert space and optimal control. SIAM J. Control Optim. 46, 356-378 (2007).
29. Meyer-Brandis, T., Øksendal, B., Zhou, X. Y.: A mean-field stochastic maximum principle via Malliavin calculus. Stoch. Int. J. Probab. Stoch. Process. 84(5-6), 643-666 (2012).
30. Veretennikov, A. Y.: On ergodic measures for McKean-Vlasov stochastic equations. In: From Stochastic Calculus to Mathematical Finance, pp. 623-633. Springer, Berlin (2006).
31. Buckdahn, R., Djehiche, B., Li, J.: A general stochastic maximum principle for SDEs of mean-field type. Appl. Math. Optim. 64, 197-216 (2011).
32. Shi, J.: Sufficient conditions of optimality for mean-field stochastic control problems. In: 12th International Conference on Control, Automation, Robotics & Vision, Guangzhou, P. R. China, December 5-7, pp. 747-752 (2012).
33. Andersson, D., Djehiche, B.: A maximum principle for SDEs of mean-field type. Appl. Math. Optim. 63, 341-356 (2011).
34. Li, J.: Stochastic maximum principle in the mean-field controls. Automatica 48, 366-373 (2012).
35. Yong, J.: A linear-quadratic optimal control problem for mean-field stochastic differential equations (2011). arXiv: 1110. 1564v1 [math. OC].
36. Yong, J., Zhou, X. Y.: Stochastic Controls. Hamiltonian Systems and HJB Equations. Springer, New York (1999).
37. Clarke, F. H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983).