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A sparse Markov chain approximation of LQ-type stochastic control problems
On the convergence of the Sakawa-Shindo algorithm in stochastic control
1. | INRIA-Saclay and Centre de Mathématiques Appliquées, Ecole Polytechnique and Laboratoire de Finance des Marchés d'Énergie, 91128 Palaiseau, France |
2. | CIFASIS - Centro Internacional Franco Argentino, de Ciencias de la Información y de Sistemas, CONICET - UNR - AMU, S2000EZP Rosario, Argentina |
3. | Institut de recherche XLIM-DMI, UMR-CNRS 7252, Faculté des sciences et techniques, Université de Limoges, 87060 Limoges, France |
References:
[1] |
J. Backhoff and F. J. Silva, Sensitivity results in stochastic optimal control: A Lagrangian perspective, ESAIM: COCV, to appear.
doi: 10.1051/cocv/2015039. |
[2] |
A. Bensoussan, Lectures on Stochastic Control, Lectures notes in Maths. Vol. 972, Springer-Verlag, Berlin, 1982. |
[3] |
A. Bensoussan, Stochastic maximum principle for distributed parameter systems, J. Franklin Inst., 315 (1983), 387-406,
doi: 10.1016/0016-0032(83)90059-5. |
[4] |
J.-M. Bismut, Linear quadratic optimal stochastic control with random coefficients, SIAM J. Control Optimization, 14 (1976), 419-444.
doi: 10.1137/0314028. |
[5] |
J.-M. Bismut, Conjugate convex functions in optimal stochastic control, J. Math. Anal. Appl., 44 (1973), 384-404.
doi: 10.1016/0022-247X(73)90066-8. |
[6] |
J.-M. Bismut, An introductory approach to duality in optimal stochastic control, SIAM Rev., 20 (1978), 62-78.
doi: 10.1137/1020004. |
[7] |
J. F. Bonnans, On an algorithm for optimal control using Pontryagin's maximum principle, SIAM J. Control Optim., 24 (1986), 579-588.
doi: 10.1137/0324034. |
[8] |
J. F. Bonnans and F. J. Silva, First and second order necessary conditions for stochastic optimal control problems, Appl. Math. Optim., 65 (2012), 403-439.
doi: 10.1007/s00245-012-9162-4. |
[9] |
A. Cadenillas and I. Karatzas, The stochastic maximum principle for linear convex optimal control with random coefficients, SIAM J. Control Optim., 33 (1995), 590-624.
doi: 10.1137/S0363012992240722. |
[10] |
A. Goldstein, Convex programming in Hilbert space, Bull. Amer. Math. Soc., 70 (1964), 709-710.
doi: 10.1090/S0002-9904-1964-11178-2. |
[11] |
U. G. Haussmann, Some examples of optimal stochastic controls or: The stochastic maximum principle at work, SIAM Rev., 23 (1981), 292-307.
doi: 10.1137/1023062. |
[12] |
H. J. Kushner, Necessary conditions for continuous parameter stochastic optimization problems, SIAM J. Control, 10 (1972), 550-565.
doi: 10.1137/0310041. |
[13] |
H. J. Kushner, On the stochastic maximum principle: Fixed time of control, J. Math. Anal. Appl., 11 (1965), 78-92.
doi: 10.1016/0022-247X(65)90070-3. |
[14] |
H. J. Kushner and F. C. Schweppe, A maximum principle for stochastic control systems, J. Math. Anal. Appl., 8 (1964), 287-302.
doi: 10.1016/0022-247X(64)90070-8. |
[15] |
L. Mazliak, An algorithm for solving a stochastic control problem, Stochastic analysis and applications, 14 (1996), 513-533.
doi: 10.1080/07362999608809455. |
[16] |
L. Mou and J. Yong, A variational formula for stochastic controls and some applications, Pure Appl. Math. Q., 3 (2007), 539-567.
doi: 10.4310/PAMQ.2007.v3.n2.a7. |
[17] |
S. G. Peng, A general stochastic maximum principle for optimal control problems, SIAM J. Control Optim., 28 (1990), 966-979.
doi: 10.1137/0328054. |
[18] |
L. Pontryagin, V. Boltyanskiĭ, R. Gamkrelidze and E. Mishchenko, The Mathematical Theory of Optimal Processes, Gordon & Breach Science Publishers, New York, 1986, Reprint of the 1962 English translation. |
[19] |
Y. Sakawa and Y. Shindo, On global convergence of an algorithm for optimal control, IEEE Trans. Automat. Control, 25 (1980), 1149-1153.
doi: 10.1109/TAC.1980.1102517. |
[20] |
J. Yong and X. Zhou, Stochastic Controls, Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999.
doi: 10.1007/978-1-4612-1466-3. |
show all references
References:
[1] |
J. Backhoff and F. J. Silva, Sensitivity results in stochastic optimal control: A Lagrangian perspective, ESAIM: COCV, to appear.
doi: 10.1051/cocv/2015039. |
[2] |
A. Bensoussan, Lectures on Stochastic Control, Lectures notes in Maths. Vol. 972, Springer-Verlag, Berlin, 1982. |
[3] |
A. Bensoussan, Stochastic maximum principle for distributed parameter systems, J. Franklin Inst., 315 (1983), 387-406,
doi: 10.1016/0016-0032(83)90059-5. |
[4] |
J.-M. Bismut, Linear quadratic optimal stochastic control with random coefficients, SIAM J. Control Optimization, 14 (1976), 419-444.
doi: 10.1137/0314028. |
[5] |
J.-M. Bismut, Conjugate convex functions in optimal stochastic control, J. Math. Anal. Appl., 44 (1973), 384-404.
doi: 10.1016/0022-247X(73)90066-8. |
[6] |
J.-M. Bismut, An introductory approach to duality in optimal stochastic control, SIAM Rev., 20 (1978), 62-78.
doi: 10.1137/1020004. |
[7] |
J. F. Bonnans, On an algorithm for optimal control using Pontryagin's maximum principle, SIAM J. Control Optim., 24 (1986), 579-588.
doi: 10.1137/0324034. |
[8] |
J. F. Bonnans and F. J. Silva, First and second order necessary conditions for stochastic optimal control problems, Appl. Math. Optim., 65 (2012), 403-439.
doi: 10.1007/s00245-012-9162-4. |
[9] |
A. Cadenillas and I. Karatzas, The stochastic maximum principle for linear convex optimal control with random coefficients, SIAM J. Control Optim., 33 (1995), 590-624.
doi: 10.1137/S0363012992240722. |
[10] |
A. Goldstein, Convex programming in Hilbert space, Bull. Amer. Math. Soc., 70 (1964), 709-710.
doi: 10.1090/S0002-9904-1964-11178-2. |
[11] |
U. G. Haussmann, Some examples of optimal stochastic controls or: The stochastic maximum principle at work, SIAM Rev., 23 (1981), 292-307.
doi: 10.1137/1023062. |
[12] |
H. J. Kushner, Necessary conditions for continuous parameter stochastic optimization problems, SIAM J. Control, 10 (1972), 550-565.
doi: 10.1137/0310041. |
[13] |
H. J. Kushner, On the stochastic maximum principle: Fixed time of control, J. Math. Anal. Appl., 11 (1965), 78-92.
doi: 10.1016/0022-247X(65)90070-3. |
[14] |
H. J. Kushner and F. C. Schweppe, A maximum principle for stochastic control systems, J. Math. Anal. Appl., 8 (1964), 287-302.
doi: 10.1016/0022-247X(64)90070-8. |
[15] |
L. Mazliak, An algorithm for solving a stochastic control problem, Stochastic analysis and applications, 14 (1996), 513-533.
doi: 10.1080/07362999608809455. |
[16] |
L. Mou and J. Yong, A variational formula for stochastic controls and some applications, Pure Appl. Math. Q., 3 (2007), 539-567.
doi: 10.4310/PAMQ.2007.v3.n2.a7. |
[17] |
S. G. Peng, A general stochastic maximum principle for optimal control problems, SIAM J. Control Optim., 28 (1990), 966-979.
doi: 10.1137/0328054. |
[18] |
L. Pontryagin, V. Boltyanskiĭ, R. Gamkrelidze and E. Mishchenko, The Mathematical Theory of Optimal Processes, Gordon & Breach Science Publishers, New York, 1986, Reprint of the 1962 English translation. |
[19] |
Y. Sakawa and Y. Shindo, On global convergence of an algorithm for optimal control, IEEE Trans. Automat. Control, 25 (1980), 1149-1153.
doi: 10.1109/TAC.1980.1102517. |
[20] |
J. Yong and X. Zhou, Stochastic Controls, Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999.
doi: 10.1007/978-1-4612-1466-3. |
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