Annals of Probability

Nonlinear Kolmogorov equations in infinite dimensional spaces: the backward stochastic differential equations approach and applications to optimal control

Marco Fuhrman

Full-text: Open access

Abstract

Solutions of semilinear parabolic differential equations in infinite dimensional spaces are obtained by means of forward and backward infinite dimensional stochastic evolution equations. Parabolic equations are intended in a mild sense that reveals to be suitable also towards applications to optimal control.

Article information

Source
Ann. Probab., Volume 30, Number 3 (2002), 1397-1465.

Dates
First available in Project Euclid: 20 August 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1029867132

Digital Object Identifier
doi:10.1214/aop/1029867132

Mathematical Reviews number (MathSciNet)
MR1920272

Zentralblatt MATH identifier
1017.60076

Subjects
Primary: 60H30: Applications of stochastic analysis (to PDE, etc.) 35R15: Partial differential equations on infinite-dimensional (e.g. function) spaces (= PDE in infinitely many variables) [See also 46Gxx, 58D25]
Secondary: 93E20: Optimal stochastic control 49L20: Dynamic programming method

Keywords
Backward stochastic differential equations partial differential equations in infinite dimensional spaces Hamilton--Jacobi--Bellman equation, stochastic optimal control

Citation

Fuhrman, Marco. Nonlinear Kolmogorov equations in infinite dimensional spaces: the backward stochastic differential equations approach and applications to optimal control. Ann. Probab. 30 (2002), no. 3, 1397--1465. doi:10.1214/aop/1029867132. https://projecteuclid.org/euclid.aop/1029867132


Export citation

References

  • [1] AMBROSETTI, A. and PRODI, G. (1995). A Primer of Nonlinear Analy sis. Cambridge Univ. Press.
  • [2] BARBU, V. and DA PRATO, G. (1983). Hamilton-Jacobi Equations in Hilbert Spaces. Longman, Essex.
  • [3] BONACCORSI, S. (1998). Some applications in Malliavin calculus. Ph.D. thesis, Dept. Mathematics, Univ. Trento.
  • [4] BONACCORSI, S. and FUHRMAN, M. (1999). Regularity results for infinite dimensional diffusions. A Malliavin calculus approach. Rend. Mat. Acc. Lincei 10 35-45.
  • [5] CANNARSA, P. and DA PRATO, G. (1991). Second-order Hamilton-Jacobi equations in infinite dimensions. SIAM J. Control Optim. 29 474-492.
  • [6] CANNARSA, P. and DA PRATO, G. (1992). Direct solution of a second-order Hamilton-Jacobi equations in Hilbert spaces. In Stochastic Partial Differential Equations and Applications (G. Da Prato and L. Tubaro, eds.) 72-85. Longman, Essex.
  • [7] CRANDALL, M. G., ISHII, H. and LIONS, P. L. (1992). User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27 1-67.
  • [8] CRANDALL, M. G., KOCAN, M. and ´SWI ¸ECH, A. (1993/94). On partial sup-convolutions, a lemma of P. L. Lions and viscosity solutions in Hilbert spaces. Adv. Math. Sci. Appl. 3 (Special Issue) 1-15.
  • [9] DA PRATO, G. and ZABCZy K, J. (1992). Stochastic equations in infinite dimensions. In Ency clopedia of Mathematics and Its Applications 44. Cambridge Univ. Press.
  • [10] DA PRATO, G. and ZABCZy K, J. (1996). Ergodicity for Infinite-Dimensional Sy stems. Cambridge Univ. Press.
  • [11] EL KAROUI, N. (1997). Backward stochastic differential equations a general introduction. In Backward Stochastic Differential Equations (N. El Karoui and L. Mazliak, eds.) 7-26. Longman, Essex.
  • [12] FLEMING, W. H. and SONER, H. M. (1993). Controlled Markov Processes and Viscosity Solutions. Springer, New York.
  • [13] FUHRMAN, M. (1996). Smoothing properties of transition semigroups in Hilbert spaces. NoDEA 3 445-464.
  • [14] FUHRMAN, M. and TESSITORE, G. (2001). The Bismut-Elworthy formula for Backward SDE's and applications to nonlinear Kolmogorov equations and control in infinite dimensional spaces. Preprint, Dipartimento di Matematica Politecnico di Milano, 470/P 1-25.
  • [15] GOLDy S, B. and GOZZI, F. Second order Hamilton-Jacobi equations in Hilbert spaces and stochastic control: an approach via invariant measures. Unpublished manuscript.
  • [16] GOZZI, F. (1995). Regularity of solutions of second order Hamilton-Jacobi equations and application to a control problem. Comm. Partial Differential Equations 20 775-826.
  • [17] GOZZI, F. (1996). Global regular solutions of second order Hamilton-Jacobi equations in Hilbert spaces with locally Lipschitz nonlinearities. J. Math. Anal. Appl. 198 399-443.
  • [18] GOZZI, F., ROUY, E. and ´SWI ¸ECH, A. (2000). Second order Hamilton-Jacobi equations in Hilbert spaces and stochastic boundary control. SIAM J. Control Optim. 38 400-430.
  • [19] GOZZI, F. and ´SWI ¸ECH, A. (2000). Hamilton-Jacobi-Bellman equations for the optimal control of the Duncan-Mortensen-Zakai equation. J. Funct. Anal. 172 466-510.
  • [20] GRORUD, A. and PARDOUX, E. (1992). Intégrales Hilbertiennes anticipantes par rapport à un processus de Wiener cy lindrique et calcul stochastique associé. Appl. Math. Optim. 25 31-49.
  • [21] KOCAN, M. and ´SWI ¸ECH, A. (1995). Second order unbounded parabolic equations in separated form. Studia Math. 115 291-310.
  • [22] HENRY, D. (1981). Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math. 840. Springer, Berlin.
  • [23] HU, Y. and PENG, S. (1991). Adapted solution of a backward semilinear stochastic evolution equation. Stochastic Anal. Appl. 9 445-459.
  • [24] LEÓN, J. A. and NUALART, D. (1998). Stochastic evolution equations with random generators. Ann. Probab. 26 149-186.
  • [25] LIONS, P. L. (1988). Viscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions. I. The case of bounded stochastic evolutions. Acta Math. 161 243-278.
  • [26] LIONS, P. L. (1989). Viscosity solutions of fully nonlinear second order equations and optimal stochastic control in infinite dimensions. II. Optimal control of Zakai's equation. Stochastic Partial Differential Equations and Applications II. Lecture Notes in Math. 1390 147-170. Springer, Berlin.
  • [27] LIONS, P. L. (1989). 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 1-18.
  • [28] MA, J. and YONG, J. (1997). Adapted solution of a degenerate backward SPDE with applications. Stochastic Process. Appl. 70 59-84.
  • [29] MA, J. and YONG, J. (1999). Forward-Backward Stochastic Differential Equations and Their Applications. Lecture Notes in Math. 1702. Springer, Berlin.
  • [30] MA, Z. M. and RÖCKNER, M. (1992). Introduction to the Theory of (Non-Sy mmetric) Dirichlet Forms. Springer, Berlin.
  • [31] MUSIELA, M. (1993). Stochastic PDEs and term structure models. J. Internationale de Finance. IGR-AFFI, La Baule.
  • [32] NUALART, D. (1995). The Malliavin calculus and related topics. In Probability and Its Applications. Springer, Berlin.
  • [33] NUALART, D. and PARDOUX, E. (1988). Stochastic calculus with anticipative integrands. Probab. Theory Related Fields 78 535-581.
  • [34] PARDOUX, E. (1998). BSDEs and viscosity solutions of a sy stem of semilinear parabolic and elliptic PDEs of second order. In Stochastic Analy sis and Related Topics: The Geilo Workshop 1996 (L. Decreusefond, J. Gjerde, B. Øksendal and A. S. Üstünel, eds.) 76-127. Birkhäuser, Berlin.
  • [35] PARDOUX, E. and PENG, S. (1990). Adapted solution of a backward stochastic differential equation. Sy stems and Control Lett. 14 55-61.
  • [36] PARDOUX, E. and PENG, S. (1992). Backward stochastic differential equations and quasilinear parabolic partial differential equations. Stochastic Partial Differential Equations and Their Applications. Lecture Notes in Control Inf. Sci. 176 200-217. Springer, Berlin.
  • [37] RÖCKNER, M. (1999). Lp-analysis of finite and infinite dimensional diffusion operators. Stochastic PDE's and Kolmogorov Equations in Infinite Dimensions. Lecture Notes in Math. 1715 65-116. Springer, Berlin.
  • [38] ´SWI ¸ECH, A. (1993). Viscosity solutions of fully nonlinear partial differential equations with "unbounded" terms in infinite dimensions. Ph.D. dissertation, Univ. California, Santa Barbara.
  • [39] ´SWI ¸ECH, A. (1994). "Unbounded" second order partial differential equations in infinitedimensional Hilbert spaces. Comm. Partial Differential Equations 19 1999-2036.
  • [40] TESSITORE, G. (1996). Existence, uniqueness and space regularity of the adapted solutions of a backward SPDE. Stochastic Anal. Appl. 14 461-486.
  • [41] ZABCZy K, J. (1999). Parabolic equations on Hilbert spaces. Stochastic PDE's and Kolmogorov Equations in Infinite Dimensions. Lecture Notes in Math. 1715 117-213. Springer, Berlin.
  • [42] ZABCZy K, J. (2000). Stochastic invariance and consistency of financial models. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 11 67-80.
  • PIAZZA LEONARDO DA VINCI, 32 20133 MILANO ITALY E-MAIL: marco.fuhrman@polimi.it DIPARTIMENTO DI MATEMATICA UNIVERSITÁ DI PARMA VIA D'AZEGLIO, 85 43100 PARMA ITALY E-MAIL: gianmario.tessitore@unipr.it