nach oben

Erschienen in:

2021 | OriginalPaper | Buchkapitel

HJB Equation, Dynamic Programming Principle, and Stochastic Optimal Control

verfasst von : Andrzej Święch

Erschienen in: Nonlinear Partial Differential Equations for Future Applications

Verlag: Springer Singapore

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

The paper is an extended version of lecture notes from a mini-course given by the author in the workshop Optimal Control and PDE in Tohoku University in 2017. The main objective of the lecture notes is to give a short but rigorous introduction to the dynamic programming approach to stochastic optimal control problems. The manuscript discusses, among other things, the classical necessary and sufficient conditions for optimality, properties of the value function, and it contains a proof of the dynamic programming principle, and a proof that the value function is a unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Vorheriges Kapitel Singular Limit Problem to the Keller-Segel System in Critical Spaces and Related Medical Problems—An Application of Maximal Regularity

Nächstes Kapitel Regularity of Solutions of Obstacle Problems –Old & New–

Bardi, M., Capuzzo-Dolcetta, I.: Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, with appendices by Maurizio Falcone and Pierpaolo Soravia. In: Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston, MA (1997)

Bensoussan, A., Lions, J.-L.: Applications of variational inequalities in stochastic control, Translated from the French. In: Studies in Mathematics and its Applications, 12. North-Holland Publishing Co., Amsterdam-New York (1982)

Borkar, V.S.: Optimal Control of Diffusion Processes, Pitman Research Notes in Mathematics Series, vol. 203, Longman Scientific & Technical, Harlow; copublished in the United States with Wiley, Inc., New York (1989)

Bouchard, B., Touzi, N.: Weak dynamic programming principle for viscosity solutions. SIAM J. Control Optim. 49(3), 948–962 (2011)MathSciNetCrossRef

Claisse, J., Talay, D., Tan, X.: A pseudo-Markov property for controlled diffusion processes. SIAM J. Control Optim. 54(2), 1017–1029 (2016)MathSciNetCrossRef

Crandall, M.G., Ishii, H., Lions, P.L.: User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27(1), 1–67 (1992)

El Karoui, N.: Les aspects probabilistes du contrôle stochastique. (French) [The probabilistic aspects of stochastic control] Ninth Saint Flour Probability Summer School-1979 (Saint Flour, 1979), pp. 73–238, Lecture Notes in Mathematics, 876, Springer, Berlin-New York (1981)

El Karoui, N., Nguyen, N.D., Jeanblanc-Picqué, M.: Compactification methods in the control of degenerate diffusions: existence of an optimal control. Stochastics 20(3), 169–219 (1987)

Fabbri, G., Gozzi, F., Święch, A.: Stochastic optimal control in infinite dimension, Dynamic programming and HJB equations, with a contribution by Marco Fuhrman and Gianmario Tessitore. In: Probability Theory and Stochastic Modelling, vol. 82. Springer, Cham (2017)

10.

Fleming, W.H., Rishel, R.W.: Deterministic and stochastic optimal control. In: Applications of Mathematics, vol. 1. Springer, Berlin-New York (1975)

11.

Fleming, W.H., Soner, H.M.: Controlled Markov processes and viscosity solutions. In: Stochastic Modelling and Applied Probability, vol. 25, 2nd edn. Springer, New York (2006)

12.

Haussmann, U.G., Lepeltier, J.-P.: On the existence of optimal controls. SIAM J. Control Optim. 28(4), 851–902 (1990)MathSciNetCrossRef

13.

Krylov, N.V.: Controlled Diffusion Processes, Translated from the Russian by A. B. Aries. Applications of Mathematics, vol. 14. Springer, New York-Berlin (1980)

14.

Lions, P.-L.: Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations, I. The dynamic programming principle and applications. Comm. Partial Differ. Equ. 8(10), 1101–1174 (1983)

15.

Lions, P.-L.: Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations, II. Viscosity solutions and uniqueness. Comm. Partial Differ. Equ. 8(11), 1229–1276 (1983)

16.

Morimoto, H.: Stochastic control and mathematical modeling. In: Applications in Economics, Encyclopedia of Mathematics and its Applications, vol. 131. Cambridge University Press, Cambridge (2010)

17.

Nisio, M.: Some remarks on stochastic optimal controls. In: Proceedings of the Third Japan-USSR Symposium on Probability Theory (Tashkent, 1975). Lecture Notes in Mathematics, vol. 550, pp. 446–460. Springer, Berlin (1975)

18.

Nisio, M.: Lectures on Stochastic Control Theory, ISI Lecture Notes, 9. Macmillan Co. of India Ltd, New Delhi (1981)MATH

19.

Nisio, M.: Stochastic control theory, dynamic programming principle. In: Probability Theory and Stochastic Modelling, vol. 72, 2nd edn. Springer, Tokyo (2015)

20.

Ondreját, M.: Uniqueness for stochastic evolution equations in Banach spaces, Dissertationes Mathematicae (Rozprawy Mat.) 426 (2004), 63 pp

21.

Pham, H.: Continuous-time stochastic control and optimization with financial applications. In: Stochastic Modelling and Applied Probability, vol. 61. Springer, Berlin (2009)

22.

Soner, H.M., Touzi, N.: Dynamic programming for stochastic target problems and geometric flows. J. Eur. Math. Soc. (JEMS) 4(3), 201–236 (2002)MathSciNetCrossRef

23.

Touzi, N.: Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE, with Chapter 13 by Angès Tourin, Fields Institute Monographs, vol. 29. Springer, New York; Fields Institute for Research in Mathematical Sciences, Toronto, ON (2013)

24.

Yong, Y., Zhou, X.Y.: Stochastic controls, Hamiltonian systems and HJB equations. In: Applications of Mathematics (New York), vol. 43. Springer, New York (1999)

25.

\(\check{\rm Z}\)itković, G.: Dynamic programming for controlled Markov families: abstractly and over martingale measures. SIAM J. Control Optim. 52(3), 1597–1621 (2014)

Titel: HJB Equation, Dynamic Programming Principle, and Stochastic Optimal Control
verfasst von: Andrzej Święch
Verlag: Springer Singapore
Buch: Nonlinear Partial Differential Equations for Future Applications
Print ISBN: 978-981-334-821-9

Electronic ISBN: 978-981-334-822-6

Copyright-Jahr: 2021
DOI: https://doi.org/10.1007/978-981-33-4822-6_5

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"