Differentiability of the Feynman-Kac semigroup and a control application

Giuseppe Da Prato; Jerzy Zabczyk

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1997)

  • Volume: 8, Issue: 3, page 183-188
  • ISSN: 1120-6330

Abstract

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The Hamilton-Jacobi-Bellman equation corresponding to a large class of distributed control problems is reduced to a linear parabolic equation having a regular solution. A formula for the first derivative is obtained.

How to cite

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Da Prato, Giuseppe, and Zabczyk, Jerzy. "Differentiability of the Feynman-Kac semigroup and a control application." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 8.3 (1997): 183-188. <http://eudml.org/doc/244107>.

@article{DaPrato1997,
abstract = {The Hamilton-Jacobi-Bellman equation corresponding to a large class of distributed control problems is reduced to a linear parabolic equation having a regular solution. A formula for the first derivative is obtained.},
author = {Da Prato, Giuseppe, Zabczyk, Jerzy},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Stochastic control problem; Feynman-Kac formula; Hamilton-Jacobi equations; Hamilton-Jacobi-Bellman equation; linear parabolic equation; regular solution},
language = {eng},
month = {10},
number = {3},
pages = {183-188},
publisher = {Accademia Nazionale dei Lincei},
title = {Differentiability of the Feynman-Kac semigroup and a control application},
url = {http://eudml.org/doc/244107},
volume = {8},
year = {1997},
}

TY - JOUR
AU - Da Prato, Giuseppe
AU - Zabczyk, Jerzy
TI - Differentiability of the Feynman-Kac semigroup and a control application
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1997/10//
PB - Accademia Nazionale dei Lincei
VL - 8
IS - 3
SP - 183
EP - 188
AB - The Hamilton-Jacobi-Bellman equation corresponding to a large class of distributed control problems is reduced to a linear parabolic equation having a regular solution. A formula for the first derivative is obtained.
LA - eng
KW - Stochastic control problem; Feynman-Kac formula; Hamilton-Jacobi equations; Hamilton-Jacobi-Bellman equation; linear parabolic equation; regular solution
UR - http://eudml.org/doc/244107
ER -

References

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  1. CANNARSA, P. - DA PRATO, G., Some results on nonlinear optimal control problems and Hamilton-Jacobi equations infinite dimensions. J. Funct. Anal., 90, 1990, 27-47. Zbl0717.49022MR1047576DOI10.1016/0022-1236(90)90079-Z
  2. CANNARSA, P. - DA PRATO, G., Direct solution of a second order Hamilton-Jacobi equation in Hilbert spaces. In: G. DA PRATO - L. TUBARO (eds.), Stochastic partial differential equations and applications. PitmanResearch Notes in Mathematics Series n. 268, 1992, 72-85. Zbl0805.49016MR1222689
  3. DA PRATO, G. - DEBUSSCHE, A., Control of the stochastic Burgers model of turbulence. Scuola Normale Superiore preprint n. 4, Pisa1996. Zbl1111.49302MR1691934DOI10.1137/S0363012996311307
  4. DA PRATO, G. - ZABCZYK, J., Ergodicity for infinite dimensions. Enciclopedia of Mathematics and its Applications, Cambridge University Press, 1996. Zbl0761.60052MR1417491DOI10.1017/CBO9780511662829
  5. ELWORTHY, K. D., Stochastic flows on Riemannian manifolds. In: M. A. PINSKY - V. VIHSTUTZ (eds.), Diffusion Processes and Related Problems in Analysis. Birkhäuser, 1992, vol. II, 33-72. Zbl0758.58035MR1187985
  6. GOZZI, F., Regularity of solutions of a second order Hamilton-Jacobi equation and application to a control problem. Commun, in partial differential equations, 20 (5&6), 1995, 775-826. Zbl0842.49021MR1326907DOI10.1080/03605309508821115
  7. GOZZI, F., Global regular solutions of second order Hamilton-Jacobi equations in Hilbert spaces with locally Lipschitz nonlinearities. Journal of Mathematical Analysis and Applications, 198, 1996, 399-443. Zbl0858.35129MR1376272DOI10.1006/jmaa.1996.0090
  8. GOZZI, F. - ROUY, E., Regular solutions of second order stationary Hamilton-Jacobi equations. J. Differential Equations, to appear. Zbl0864.34058MR1409030DOI10.1006/jdeq.1996.0139
  9. LIONS, P. L., Viscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions. Part I: The case of bounded stochastic evolution. Acta Math., 161, 1988, 243-278. Zbl0757.93082MR971797DOI10.1007/BF02392299
  10. LIONS, P. L., Viscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions. Part II: Optimal control fo Zakai's equation. In: G. DA PRATO - L. TUBARO (eds.), Stochastic Partial Differential Equations and Applications. Lecture Notes in Mathematics No. 1390, Springer-Verlag, 1989, 147-170. Zbl0757.93083MR1019600DOI10.1007/BFb0083943
  11. LIONS, P. L., Viscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions. Part III: Uniqueness of viscosity solutions for general second order equations. J. Funct. Anal., 86, 1989, 1-18. Zbl0757.93084MR1013931DOI10.1016/0022-1236(89)90062-1
  12. PESZAT, S. - ZABCZYK, J., Strong Feller property and irreducibility for diffusions on Hilbert spaces. Annals of Probability, 1996. Zbl0831.60083MR1330765
  13. SWIECH, A., Viscosity solutions of fully nonlinear partial differential equations with «unbounded» terms in infinite dimensions. Ph. D. Thesis, University of California at Santa Barbara, 1993. MR2690118

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