A class of stochastic optimal control problems in Hilbert spaces: BSDEs and optimal control laws, state constraints, conditioned processes

Stochastic Processes and their Applications

Volumn 108, Issue 2, 2003, Pages 263-298

  Fuhrman, Marco ^a  

^a POLITECNICO DI MILANO   (Italy)

Author keywords

Forward backward stochastic differential equations; Stochastic optimal control

Indexed keywords

EID: 0142087747     PISSN: 03044149     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.spa.2003.09.002     Document Type: Article

References (39)

1. Barbu V. Da Prato G. Hamilton-Jacobi equations in Hilbert spaces Pitman Research Notes in Mathematics Vol. 86 1983 Pitman London
2. Cannarsa P. Da Prato G. Second-order Hamilton-Jacobi equations in infinite dimensions SIAM J. Control Optim. 29 2 1991 474-492
3. Cannarsa P. Da Prato G. Direct solution of a second-order Hamilton-Jacobi equations in Hilbert spaces Da Prato G. Tubaro L. Stochastic Partial Differential Equations and Applications Pitman Research Notes in Mathematics Vol. 268 1992 72-85 Pitman London
4. Crandall M.G. Kocan M. Świȩch A. On partial sup-convolutions, a lemma of P.L. Lions and viscosity solutions in Hilbert spaces Adv. Math. Sci. Appl. 3 Special Issue 1993/94 1-15
5. ADai Pra P. A stochastic control approach to reciprocal diffusion processes Appl. Math. Optim. 23 1991 313-329
6. Da Prato G. Zabczyk J. Stochastic equations in infinite dimensions Encyclopedia of Mathematics and its Applications Vol. 44 1992 Cambridge University Press Cambridge
7. Da Prato G. Zabczyk J. Ergodicity for infinite-dimensional systems London Mathematical Society Lecture Note Series Vol. 229 1996 Cambridge University Press Cambridge
8. Da Prato G. Zabczyk J. Differentiability of the Feynman-Kac semigroup and a control application Rend. Mat. Accad. Lincei 8 s. 9 1997 183-188
9. Da Prato, G., Zabczyk, J., 2002. Second order partial differential equations in Hilbert spaces. London Mathematical Society Lecture Note Series, Vol. 293. Cambridge University Press, Cambridge.
10. Dellacherie G. Meyer P.A. Probabilities and potential. B. Theory of martingales North-Holland Mathematics Studies Vol. 72 1982 North-Holland Amsterdam
11. El Karoui, N., Mazliak, L. (Eds.), 1997. Backward stochastic differential equations. Pitman Research Notes in Mathematics Series, Vol. 364, Longman.
12. El Karoui N. Peng S. Quenez M.C. Backward stochastic differential equations in finance Math. Finance 7 1 1997 1-71
13. Fleming W.H. Sheu S.J. Stochastic variational formula for fundamental solutions of parabolic PDE Appl. Math. Optim. 13 1985 193-204
14. Fleming W.H. Soner H.M. Controlled Markov processes and viscosity solutions Applications of Mathematics Vol. 25 1993 Springer New York
15. Fuhrman M. Tessitore G. Nonlinear Kolmogorov equations in infinite dimensional spaces: the backward stochastic differential equations approach and applications to optimal control Ann. Probab. 30 3 2002 1397-1465
16. Fuhrman M. Tessitore G. The Bismut-Elworthy formula for Backward SDE's and applications to nonlinear Kolmogorov equations and control in infinite dimensional spaces Stochastics Stochastics Rep. 74 1-2 2002 429-464
17. Fuhrman, M., Tessitore, G., 2002c. Infinite horizon backward stochastic differential equations and elliptic equations in Hilbert spaces. Preprint Dipartimento di Matematica del Politecnico di Milano, 498/P, February.
18. Gozzi F. Regularity of solutions of second order Hamilton-Jacobi equations and application to a control problem Comm. Partial Differential Equations 20 1995 775-826
19. Gozzi F. Global regular solutions of second order Hamilton-Jacobi equations in Hilbert spaces with locally Lipschitz nonlinearities J. Math. Anal. Appl. 198 1996 399-443
20. Gozzi F. Rouy E. Świȩch A. Second order Hamilton-Jacobi equations in Hilbert spaces and stochastic boundary control SIAM J. Control Optim. 38 2 2000 400-430
21. Kobylanski M. Backward stochastic differential equations and partial differential equations with quadratic growth Ann. Probab. 28 2 2000 558-602
22. Lenglart E. Transformation des martingales locales par changement absolument continu de probabilités Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 39 1 1977 65-70
23. León J.A. Nualart D. Stochastic evolution equations with random generators Ann. Probab. 26 1 1998 149-186
24. Lepeltier J.-P. San Martín J. Existence for BSDE with superlinear-quadratic coefficient Stochastics Stochastics Rep. 63 3-4 1998 227-240
25. Lions P.L. Viscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions. I. The case of bounded stochastic evolutions Acta Math. 161 3-4 1988 243-278
26. Lions P.L. Viscosity solutions of fully nonlinear second order equations and optimal stochastic control in infinite dimensions. II. Optimal control of Zakai's equation Da Prato G. Tubaro L. Stochastic partial differential equations and applications, II Lecture Notes in Mathematics Vol. 1390 1989 147-170 Springer Berlin
27. Lions P.L. Viscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions. III. Uniqueness of viscosity solutions for general second-order equations J. Funct. Anal. 86 1 1989 1-18
28. Ma J. Yong J. Forward-backward stochastic differential equations and their applications Lecture Notes in Mathematics Vol. 1702 1999 Springer Berlin
29. Métivier, M., 1978. Stochastic integration with respect to Hilbert valued martingales, representation theorems and infinite-dimensional filtering. Measure theory applications to stochastic analysis (Proceedings of the Conference, Res. Inst. Math., Oberwolfach, 1977), Lecture Notes in Mathematics, Vol. 695. Springer, Berlin, pp. 13-25.
30. Ouvrard J.Y. Représentation de martingales vectorielles de carré intégrable à valeurs dans des espaces de Hilbert réels séparables Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 33 3 1975/76 195-208
31. Pardoux E. Peng S. Adapted solution of a backward stochastic differential equation Systems Control Lett. 14 1990 55-61
32. Peng S. A generalized dynamic programming principle and Hamilton-Jacobi-Bellman equation Stochastics Stochastics Rep. 38 1992 119-134
33. Revuz, D., Yor, M., 1999. Continuous Martingales and Brownian Motion, 3rd Edition. Grundlehren der Mathematischen Wissenschaften, Vol. 293. Springer, Berlin.
34. Rozovskii B.L. Stochastic evolution systems. Linear theory and applications to nonlinear filtering Mathematics and its Applications (Soviet Series) Vol. 35 1990 Kluwer Dordrecht
35. Simão I. A conditioned Ornstein-Uhlenbeck process on a Hilbert space Stochastic Anal. Appl. 9 1991 85-98
36. Simão I. Regular fundamental solution for a parabolic equation on an infinite-dimensional space Stochastic Anal. Appl. 11 1993 235-247
37. Simão I. Regular transition densities for infinite dimensional diffusions Stochastic Anal. Appl. 11 1993 309-336
38. Świȩch, A., 1993. Viscosity solutions of fully nonlinear partial differential equations with "unbounded" terms in infinite dimensions. Ph.D. Thesis, University of California at Santa Barbara.
39. Świȩch A. "Unbounded" second order partial differential equations in infinite-dimensional Hilbert spaces Comm. Partial Differential Equations 19 11-12 1994 1999-2036