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 & Continuous Dynamical Systems - A, 2012, 32 (1) : 223-263. doi: 10.3934/dcds.2012.32.223
References:
[1]

J. P. Aubin and H. Frankowska, "Set Valued Analysis,", Birkhäuser, (1990).   Google Scholar

[2]

X. Bao, F. Delbaen and Y. Hu, Backward SDEs with superquadratic growth,, preprint, ().  doi: 10.1007/s00440-010-0271-1.  Google Scholar

[3]

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

[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.  doi: 10.1006/jfan.1997.3229.  Google Scholar

[5]

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

[6]

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

[7]

G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions,", Encyclopedia of Mathematics and its Applications 44, (1992).   Google Scholar

[8]

G. Da Prato and J. Zabczyk, "Second Order Partial Eifferential Equations in Hilbert Spaces,", London Mathematical Society Note Series, (2002).   Google Scholar

[9]

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

[10]

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

[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.  doi: 10.1137/050633548.  Google Scholar

[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.  doi: 10.1214/aop/1029867132.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1006/jmaa.1996.0090.  Google Scholar

[15]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Mathematics, (1981).   Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

E. Pardoux, BSDE's, weak convergence and homogeneization of semilinear PDE's,, in, (1999), 503.   Google Scholar

[22]

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

[23]

E. Pardoux and S. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations,, in, (1992), 200.   Google Scholar

[24]

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

[25]

A. Richou, Numerical simulation of BSDEs with drivers of quadratic growth,, preprint, ().   Google Scholar

show all references

References:
[1]

J. P. Aubin and H. Frankowska, "Set Valued Analysis,", Birkhäuser, (1990).   Google Scholar

[2]

X. Bao, F. Delbaen and Y. Hu, Backward SDEs with superquadratic growth,, preprint, ().  doi: 10.1007/s00440-010-0271-1.  Google Scholar

[3]

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

[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.  doi: 10.1006/jfan.1997.3229.  Google Scholar

[5]

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

[6]

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

[7]

G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions,", Encyclopedia of Mathematics and its Applications 44, (1992).   Google Scholar

[8]

G. Da Prato and J. Zabczyk, "Second Order Partial Eifferential Equations in Hilbert Spaces,", London Mathematical Society Note Series, (2002).   Google Scholar

[9]

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

[10]

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

[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.  doi: 10.1137/050633548.  Google Scholar

[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.  doi: 10.1214/aop/1029867132.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1006/jmaa.1996.0090.  Google Scholar

[15]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Mathematics, (1981).   Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

E. Pardoux, BSDE's, weak convergence and homogeneization of semilinear PDE's,, in, (1999), 503.   Google Scholar

[22]

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

[23]

E. Pardoux and S. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations,, in, (1992), 200.   Google Scholar

[24]

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

[25]

A. Richou, Numerical simulation of BSDEs with drivers of quadratic growth,, preprint, ().   Google Scholar

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