Your browser does not support JavaScript!
http://iet.metastore.ingenta.com
1887

access icon free Mean-field formulation for the infinite-horizon mean–variance control of discrete-time linear systems with multiplicative noises

This study considers the infinite-horizon stochastic optimal control of a discounted and long-run average costs under a mean–variance trade-off performance criterion for discrete-time linear systems subject to multiplicative noises. The authors adopt a mean-field approach to tackle the problem and get an optimal control solution in terms of a set of two generalised coupled algebraic Riccati equations (GCAREs). Then, they establish sufficient conditions for the existence of the maximal solution and necessary and sufficient conditions for the existence of the mean-square stabilising solution to the GCARE. From this solution, they derive optimal control policies to the related discounted and long-run average cost problems. A numerical example illustrates the obtained results for the multi-period portfolio selection problem in which it is desired to minimise the sum of the mean–variance trade-off costs of a portfolio against a benchmark along the time.

References

    1. 1)
      • 2. Yurchenkov, A.V.: ‘Anisotropy-based controller design for linear discrete-time systems with multiplicative noise’, J. Comput. Syst. Sci. Int., 2018, 57, pp. 864873.
    2. 2)
      • 16. Zhang, H., Qi, Q., Fu, M.: ‘Optimal stabilization control for discrete-time mean-field stochastic systems’, IEEE Trans. Autom. Control, 2019, 64, pp. 11251136.
    3. 3)
      • 12. Elliott, R.J., Ni, Y.: ‘Discrete time mean-field stochastic linear quadatic optimal control problems’, Automatica, 2013, 49, (11), pp. 32223233.
    4. 4)
      • 26. Davis, M.H.A., Vinter, R.B.: ‘Stochastic modelling and control’ (Chapman and Hall, London, 1985).
    5. 5)
      • 23. Weidmann, J.: ‘Linear operators in Hilbert spaces’ (Springer-Verlag, New York, 1980).
    6. 6)
      • 11. Cui, X., Li, X., Li, D.: ‘Unified framework of mean-field formulations for optimal multi-period mean–variance portfolio selection’, IEEE Trans. Autom. Control, 2014, 59, (7), pp. 18331844.
    7. 7)
      • 21. Damm, T.: ‘Rational matrix equations in stochastic control (lecture notes in control and information sciences)’, vol. 297 (Springer-Verlag, Berlin, Heidelberg, 2004).
    8. 8)
      • 20. Damm, T., Hinrichsen, D.: ‘Newton's method for concave operators with resolvent positive derivatives in ordered banach spaces’, Linear Algebra Appli., 2003, 363, pp. 4364.
    9. 9)
      • 17. Costa, O.L.V., Fragoso, M.D., Marques, R.P.: ‘Discrete-time Markov jump linear systems’ (Springer-Verlag, London, 2005).
    10. 10)
      • 13. Huang, J., Li, X., Yong, J.: ‘A linear quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon’, Math. Control Related Fields, 2015, 5, (1), pp. 97139.
    11. 11)
      • 24. Barbieri, F., Costa, O.L.V.: ‘A mean-field formulation for the mean–variance control of discrete-time linear systems with multiplicative noises’, Int. J. Syst. Sci., 2020, 51, (10), pp. 18251846. Available at http://dx.doi.org/10.1080/00207721.2020.1780340.
    12. 12)
      • 6. Sun, Y., Kong, S., Cui, G., et al: ‘Optimal filtering for time-varying stochastic system with delay and multiplicative noise’, IEEE Access, 2019, 7, pp. 4423944246.
    13. 13)
      • 7. Dragan, V., Morozan, T., Stoica, A.M.: ‘Mathematical methods in robust control of linear stochastic systems’ (Springer-Verlag, New York, NY, USA, 2013, 3rd edn.).
    14. 14)
      • 5. Costa, O.L.V., de Paulo, W.L.: ‘Generalized coupled algebraic Riccati equations for discrete-time Markov jump with multiplicative noise systems’, Eur. J. Control, 2008, 14, pp. 391408.
    15. 15)
      • 1. Gershon, E., Shaked, U., Yaesh, I.: ‘H control and filtering of discrete-time stochastic systems with multiplicative noise’, Automatica, 2001, 37, pp. 409417.
    16. 16)
      • 15. Yong, J.: ‘Linear-quadratic optimal control problems for mean-field stochastic differential equations’, SIAM J. Control Opt., 2013, 51, pp. 28092838.
    17. 17)
      • 10. Buckdahn, R., Djehiche, B., Li, J.: ‘A general stochastic maximum principle for SDEs of mean-field type’, Appl. Math. Opt., 2011, 64, pp. 197216.
    18. 18)
      • 9. Buckdahn, R., Djehiche, B., Li, J.: ‘Mean-field backward stochastic differential equations: a limit approach’, Ann. Prob., 2009, 37, pp. 15241565.
    19. 19)
      • 25. Diamond, S., Boyd, S.: ‘CVXPY: a python-embedded modeling language for convex optimization’, J. Mach. Learn. Res., 2016, 17, (83), pp. 15.
    20. 20)
      • 22. Schaefer, H.H.: ‘Topological vector spaces’ (Springer-Verlag, Berlin, Heidelberg, New York, 1971).
    21. 21)
      • 18. Zhou, X.Y., Li, D.: ‘Continuous-time mean–variance portfolio selection: a stochastic LQ framework’, Appl. Math. Opt., 2000, 42, pp. 1933.
    22. 22)
      • 8. Li, D., Ng, W.L.: ‘Optimal dynamic portfolio selection: multi-period mean–variance formulation’, Math. Finance, 2000, 10, pp. 387406.
    23. 23)
      • 27. Sontag, E.D.: ‘Mathematical control theory’ (Springer-Verlag, New York, 1990).
    24. 24)
      • 4. Zhu, S.S., Li, D., Wang, S.Y.: ‘Risk control over bankruptcy in dynamic portfolio selection: a generalized mean–variance formulation’, IEEE Trans. Autom. Control, 2004, 49, pp. 447457.
    25. 25)
      • 19. Saberi, A., Sannuti, P.: ‘H2-optimal control’ (Prentice Hall, New Jersey, 1995).
    26. 26)
      • 14. Ni, Y.H., Elliott, R., Li, X.: ‘Discrete-time mean-field stochastic linear-quadratic optimal control problems, II: infinite horizon case’, Automatica, 2015, 57, pp. 6577.
    27. 27)
      • 3. Barbieri, F., Costa, O.L.V.: ‘Optimal control with constrained total variance for Marvov jump linear systems with multiplicative noises’, Int. J. Syst. Sci., 2018, 49, (6), pp. 11781187. Available at https://www.tandfonline.com/doi/abs/10.1080/00207721.2018.1441469.
http://iet.metastore.ingenta.com/content/journals/10.1049/iet-cta.2020.0442
Loading

Related content

content/journals/10.1049/iet-cta.2020.0442
pub_keyword,iet_inspecKeyword,pub_concept
6
6
Loading
This is a required field
Please enter a valid email address