January  2012, 32(1): 223-263. doi: 10.3934/dcds.2012.32.223

Hamilton Jacobi Bellman equations in infinite dimensions with quadratic and superquadratic Hamiltonian

1. 

Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, via Cozzi 53, 20125 Milano, Italy

Received  July 2010 Revised  April 2011 Published  September 2011

We consider Hamilton Jacobi Bellman equations in an infinite dimensional Hilbert space, with quadratic (respectively superquadratic) Hamiltonian and with continuous (respectively lipschitz continuous) final condition. This allows to study stochastic optimal control problems for suitable controlled state equations with unbounded control processes. The results are applied to controlled heat equations.
Citation: Federica Masiero. Hamilton Jacobi Bellman equations in infinite dimensions with quadratic and superquadratic Hamiltonian. Discrete and Continuous Dynamical Systems, 2012, 32 (1) : 223-263. doi: 10.3934/dcds.2012.32.223
References:
[1]

J. P. Aubin and H. Frankowska, "Set Valued Analysis," Birkhäuser, Boston, 1990.

[2]

X. Bao, F. Delbaen and Y. Hu, Backward SDEs with superquadratic growth, preprint, arXiv:0902.3316, to appear on Probability Theory and Related Fields. doi: 10.1007/s00440-010-0271-1.

[3]

P. Briand and F. Confortola, BSDEs with stochastic Lipschitz condition and quadratic PDEs in Hilbert spaces, Stochastic Process. Appl., 118 (2008), 818-838. doi: 10.1016/j.spa.2007.06.006.

[4]

P. Briand and Y. Hu, Stability of BSDEs with random terminal time and homogenization of semilinear elliptic PDEs, J. Funct. Anal., 155 (1998), 455-494 doi: 10.1006/jfan.1997.3229.

[5]

S. Cerrai, A Hille-Yosida theorem for weakly continuous semigroups, Semigroup Forum, 49 (1994), 349-367. doi: 10.1007/BF02573496.

[6]

S. Cerrai and F. Gozzi, Strong solutions of Cauchy problems associated to weakly continuous semigroups, Differential Integral Equations, 8 (1995), 465-486.

[7]

G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions," Encyclopedia of Mathematics and its Applications 44, Cambridge University Press, 1992.

[8]

G. Da Prato and J. Zabczyk, "Second Order Partial Eifferential Equations in Hilbert Spaces," London Mathematical Society Note Series, 293, Cambridge University Press, Cambridge, 2002.

[9]

W. H. Fleming and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions," Applications of Mathematics 25. Springer-Verlag, 1993.

[10]

M. Fuhrman, Smoothing properties of nonlinear stochastic equations in Hilbert spaces, NoDEA Nonlinear Differential Equations Appl., 3 (1996), 445-464.

[11]

M. Fuhrman, Y. Hu and G. Tessitore, On a class of stochastic optimal control problems related to BSDEs with quadratic growth, SIAM J. Control Optim., 45 (2006), 1279-1296. doi: 10.1137/050633548.

[12]

M. Fuhrman and G. Tessitore, Nonlinear Kolmogorov equations in infinite dimensional spaces: The backward stochastic differential equations approach and applications to optimal control, Ann. Probab., 30 (2002), 1397-1465. doi: 10.1214/aop/1029867132.

[13]

F. Gozzi, Regularity of solutions of a second order Hamilton-Jacobi equation and application to a control problem, Comm. Partial Differential Equations, 20 (1995), 775-826.

[14]

F. Gozzi, Global regular solutions of second order Hamilton-Jacobi equations in Hilbert spaces with locally Lipschitz nonlinearities, J. Math. Anal. Appl., 198 (1996), 399-443. doi: 10.1006/jmaa.1996.0090.

[15]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.

[16]

Y. Hu, P. Imkeller and M. Müller, Utility maximization in incomplete markets, Ann. Appl. Probab., 15 (2005), 1691-1712. doi: 10.1214/105051605000000188.

[17]

M. Kobylanski, Backward stochastic differential equations and partial differential equations with quadratic growth, Ann. Probab., 28 (2000), 558-602. doi: 10.1214/aop/1019160253.

[18]

J. M. Lasry and P. L. Lions, A remark on regularization in Hilbert spaces, Israel. J. Math., 55 (1986), 257-266.

[19]

F. Masiero, Semilinear Kolmogorov equations and applications to stochastic optimal control, Appl. Math. Optim., 51 (2005), 201-250. doi: 10.1007/s00245-004-0810-6.

[20]

F. Masiero, Regularizing properties for transition semigroups and semilinear parabolic equations in Banach spaces, Electron. J. Probab., 12 (2007), 387-419.

[21]

E. Pardoux, BSDE's, weak convergence and homogeneization of semilinear PDE's, in "Nonlinear Analysis, Differential Equations and Control," (eds. F.H. Clarke, R.J. Stern), Kluwer, (1999), 503-549.

[22]

E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett., 14 (1990), 55-61. doi: 10.1016/0167-6911(90)90082-6.

[23]

E. Pardoux and S. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations, in "Stochastic Partial Differential Equations and Their Applications" (eds. B.L. Rozowskii and R.B. Sowers), Lecture Notes in Control Inf. Sci. 176, Springer, (1992), 200-217.

[24]

S. Peszat and J. Zabczyk, Strong Feller property and irreducibility for diffusions on Hilbert sapces, Ann. Probab., 23 (1995), 157-172. doi: 10.1214/aop/1176988381.

[25]

A. Richou, Numerical simulation of BSDEs with drivers of quadratic growth, preprint, arXiv:1001.0401, to appear on Annals of Applied Probability.

show all references

References:
[1]

J. P. Aubin and H. Frankowska, "Set Valued Analysis," Birkhäuser, Boston, 1990.

[2]

X. Bao, F. Delbaen and Y. Hu, Backward SDEs with superquadratic growth, preprint, arXiv:0902.3316, to appear on Probability Theory and Related Fields. doi: 10.1007/s00440-010-0271-1.

[3]

P. Briand and F. Confortola, BSDEs with stochastic Lipschitz condition and quadratic PDEs in Hilbert spaces, Stochastic Process. Appl., 118 (2008), 818-838. doi: 10.1016/j.spa.2007.06.006.

[4]

P. Briand and Y. Hu, Stability of BSDEs with random terminal time and homogenization of semilinear elliptic PDEs, J. Funct. Anal., 155 (1998), 455-494 doi: 10.1006/jfan.1997.3229.

[5]

S. Cerrai, A Hille-Yosida theorem for weakly continuous semigroups, Semigroup Forum, 49 (1994), 349-367. doi: 10.1007/BF02573496.

[6]

S. Cerrai and F. Gozzi, Strong solutions of Cauchy problems associated to weakly continuous semigroups, Differential Integral Equations, 8 (1995), 465-486.

[7]

G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions," Encyclopedia of Mathematics and its Applications 44, Cambridge University Press, 1992.

[8]

G. Da Prato and J. Zabczyk, "Second Order Partial Eifferential Equations in Hilbert Spaces," London Mathematical Society Note Series, 293, Cambridge University Press, Cambridge, 2002.

[9]

W. H. Fleming and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions," Applications of Mathematics 25. Springer-Verlag, 1993.

[10]

M. Fuhrman, Smoothing properties of nonlinear stochastic equations in Hilbert spaces, NoDEA Nonlinear Differential Equations Appl., 3 (1996), 445-464.

[11]

M. Fuhrman, Y. Hu and G. Tessitore, On a class of stochastic optimal control problems related to BSDEs with quadratic growth, SIAM J. Control Optim., 45 (2006), 1279-1296. doi: 10.1137/050633548.

[12]

M. Fuhrman and G. Tessitore, Nonlinear Kolmogorov equations in infinite dimensional spaces: The backward stochastic differential equations approach and applications to optimal control, Ann. Probab., 30 (2002), 1397-1465. doi: 10.1214/aop/1029867132.

[13]

F. Gozzi, Regularity of solutions of a second order Hamilton-Jacobi equation and application to a control problem, Comm. Partial Differential Equations, 20 (1995), 775-826.

[14]

F. Gozzi, Global regular solutions of second order Hamilton-Jacobi equations in Hilbert spaces with locally Lipschitz nonlinearities, J. Math. Anal. Appl., 198 (1996), 399-443. doi: 10.1006/jmaa.1996.0090.

[15]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.

[16]

Y. Hu, P. Imkeller and M. Müller, Utility maximization in incomplete markets, Ann. Appl. Probab., 15 (2005), 1691-1712. doi: 10.1214/105051605000000188.

[17]

M. Kobylanski, Backward stochastic differential equations and partial differential equations with quadratic growth, Ann. Probab., 28 (2000), 558-602. doi: 10.1214/aop/1019160253.

[18]

J. M. Lasry and P. L. Lions, A remark on regularization in Hilbert spaces, Israel. J. Math., 55 (1986), 257-266.

[19]

F. Masiero, Semilinear Kolmogorov equations and applications to stochastic optimal control, Appl. Math. Optim., 51 (2005), 201-250. doi: 10.1007/s00245-004-0810-6.

[20]

F. Masiero, Regularizing properties for transition semigroups and semilinear parabolic equations in Banach spaces, Electron. J. Probab., 12 (2007), 387-419.

[21]

E. Pardoux, BSDE's, weak convergence and homogeneization of semilinear PDE's, in "Nonlinear Analysis, Differential Equations and Control," (eds. F.H. Clarke, R.J. Stern), Kluwer, (1999), 503-549.

[22]

E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett., 14 (1990), 55-61. doi: 10.1016/0167-6911(90)90082-6.

[23]

E. Pardoux and S. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations, in "Stochastic Partial Differential Equations and Their Applications" (eds. B.L. Rozowskii and R.B. Sowers), Lecture Notes in Control Inf. Sci. 176, Springer, (1992), 200-217.

[24]

S. Peszat and J. Zabczyk, Strong Feller property and irreducibility for diffusions on Hilbert sapces, Ann. Probab., 23 (1995), 157-172. doi: 10.1214/aop/1176988381.

[25]

A. Richou, Numerical simulation of BSDEs with drivers of quadratic growth, preprint, arXiv:1001.0401, to appear on Annals of Applied Probability.

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