Linear-quadratic optimal control problems for mean-field stochastic differential equations

SIAM Journal on Control and Optimization

Volumn 51, Issue 4, 2013, Pages 2809-2838

  Yong, Jiongmin ^a  

^a UNIVERSITY OF CENTRAL FLORIDA   (United States)

Author keywords

Feedback representation; Linear quadratic optimal control; Mean field stochastic differential equation; Riccati differential equation

Indexed keywords

DECOUPLING TECHNIQUE; FORWARD-BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS; LINEAR-QUADRATIC OPTIMAL CONTROL PROBLEMS; LINEAR-QUADRATIC OPTIMAL CONTROLS; OPTIMALITY SYSTEM; RICCATI DIFFERENTIAL EQUATION; STOCHASTIC DIFFERENTIAL EQUATIONS; VARIATIONAL METHODS;

CONTROL; OPTIMAL CONTROL SYSTEMS;

DIFFERENTIAL EQUATIONS;

EID: 84885675349     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/120892477     Document Type: Article

References (38)

1. N. U. Ahmed and X. Ding, A semilinear McKean-Vlasov stochastic evolution equation in Hilbert space, Stochastic Process. Appl., 60 (1995), pp. 65-85.
2. N. U. Ahmed and X. Ding, Controlled McKean-Vlasov equations, Commun. Appl. Anal., 5 (2001), pp. 183-206. (Pubitemid 33286366)
3. N. U. Ahmed, Nonlinear diffusion governed by McKean-Vlasov equation on Hilbert space and optimal control, SIAM J. Control Optim., 46 (2007), pp. 356-378. (Pubitemid 351271301)
4. D. Andersson and B. Djehiche, A maximum principle for SDEs of mean-field type, Appl. Math. Optim., 63 (2011), pp. 341-356.
5. R. Bellman, R. Kalaba, and G. M. Wing, Invariant imbedding and the reduction of two-point boundary value problems to initial value problems, Proc. Natl. Acad. Sci. USA, 46 (1960), pp. 1646-1649.
6. T. Björk and A. Murgoci, A General Theory of Markovian Time Inconsistent Stochastic Control Problem, working paper.
7. T. Björk, A. Murgoci, and X. Y. Zhou, Mean varaiance portfolio optimization with state dependent risk aversion, Math. Finance, in press.
8. V. S. Borkar and K. S. Kumar, McKean-Vlasov limit in portfolio optimization, Stochastic Anal. Appl., 28 (2010), pp. 884-906.
9. R. Buckdahn, B. Djehiche, and J. Li, A General Maximum Principle for SDEs of Mean-Field Type, Appl. Math. Optim., 64 (2011), pp. 197-216.
10. R. Buckdahn, B. Djehiche, J. Li, and S. Peng, Mean-field backward stochastic differential equations: A limit approach, Ann. Probab., 37 (2009), pp. 1524-1565.
11. R. Buckdahn, J. Li, and S. Peng, Mean-field backward stochastic differential equations and related partial differential equations, Stochastic Process. Appl., 119 (2009), pp. 3133-3154
12. T. Chan, Dynamics of the McKean-Vlasov equation, Ann. Probab., 22 (1994), pp. 431-441.
13. T. Chiang, McKean-Vlasov equations with discontinuous coefficients, Soochow J. Math., 20 (1994), pp. 507-526.
14. D. Crisan and J. Xiong, Approximate McKean-Vlasov representations for a class of SPDEs, Stochastics, 82 (2010), pp. 53-68.
15. D. A. Dawson, Critical dynamics and fluctuations for a mean-field model of cooperative behavior, J. Statist. Phys., 31 (1983), pp. 29-85.
16. D. A. Dawson and J. Gärtner, Large deviations from the McKean-Vlasov limit for weakly interacting diffusions, Stochastics, 20 (1987), pp. 247-308.
17. I. Ekeland and A. Lazrak, The golden rule when preferences are time inconsistent, Math. Finan. Econ., 4 (2010), pp. 29-55.
18. I. Ekeland and T. A. Pirvu, Investment and consumption without commitment, Math. Finan. Econ., 2 (2008), pp. 57-86.
19. I. Ekeland, O. Mbodji, and T. A. Pirvu, Time-Consistent Portfolio Management, SIAM J. Financial Math., 3 (2012), pp. 1-32.
20. J. Gärtner, On the McKean-Vlasov limit for interacting diffusions, Math. Nachr., 137 (1988), pp. 197-248.
21. C. Graham, McKean-Vlasov Ito-Skorohod equations, and nonlinear diffusions with discrete jump sets, Stochastic Process. Appl., 40 (1992), pp. 69-82.
22. Y. Hu, H. Jin, and X. Y. Zhou, Time-Inconsistent Stochastic Linear-Quadratic Control, arXiv:1111.0818v1, 2011.
23. M. Huang, R. P. Malhamé, and P. E. Caines, Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Commun. Inf. Systems, 6 (2006), pp. 221-252.
24. M. Kac, Foundations of kinetic theory, in Proceedings of the 3rd Berkeley Symposium on Mathematical Statistics and Probability, vol. 3, 1956, pp. 171-197.
25. P. M. Kotelenez and T. G. Kurtz, Macroscopic limit for stochastic partial differential equations of McKean-Vlasov type, Prob. Theory Related Fields, 146 (2010), pp. 189-222.
26. J. Ma, P. Protter, and J. Yong, Solving forward-backward stochastic differential equations explicitly: A four-step scheme, Probab. Theory Related Fields, 98 (1994), pp. 339-359.
27. J. Ma and J. Yong, Forward-Backward Stochastic Differential Equations and Their Applications, Lecture Notes in Math. 1702, Springer-Verlag, New York, 1999.
28. N. I. Mahmudov and M. A. McKibben, On a class of backward McKean-Vlasov stochastic equations in Hilbert space: Existence and convergence properties, Dynam. Systems Appl., 16 (2007), pp. 643-664.
29. H. P. McKean, A class of Markov processes associated with nonlinear parabolic equations, Proc. Natl. Acad. Sci. USA, 56 (1966), pp. 1907-1911.
30. T. Meyer-Brandis, B. Oksendal, and X. Zhou, A mean-field stochastic maximum principle via Malliavin calculus, Stochastics, 84 (2012), pp. 643-666.
31. L. Mou and J. Yong, Two-person zero-sum linear quadratic stochastic differential games by a Hilbert space method, J. Industrial Management Optim., 2 (2006), pp. 95-117.
32. D. Nualart, The Malliavin Calculus and Related Topics, 2nd ed., Springer-Verlag, Berlin, 2006.
33. J. Y. Park, P. Balasubramaniam, and Y. H. Kang, Controllability of McKean-Vlasov stochastic integrodifferential evolution equation in Hilbert spaces, Numer. Funct. Anal. Optim., 29 (2008), pp. 1328-1346.
34. M. Scheutzow, Uniqueness and non-uniqueness of solutions of Vlasov-McKean equations, J. Aust. Math. Soc. Ser. A, 43 (1987), pp. 246-256.
35. A. Yu. Veretennikov, On ergodic measures for McKean-Vlasov stochastic equations, in Stochastic Calculus to Mathematical Finance, Springer, Berlin, 2006, pp. 623-633.
36. J. Yong, A deterministic linear quadratic time-inconsistent optimal control problem, Math. Control Related Fields, 1 (2011), pp. 83-118.
37. J. Yong, Time-inconsistent optimal control problems and the equilibrium HJB equation, Math. Control Related Fields, 2 (2012), pp. 271-329.
38. J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999.