June  2013, 6(3): 803-824. doi: 10.3934/dcdss.2013.6.803

On backward stochastic differential equations in infinite dimensions

1. 

University of Antwerp, Department of Mathematics and Computer Science, Middelheimlaan 1, 2020 Antwerp, Belgium

Received  January 2010 Revised  November 2010 Published  December 2012

In the present paper we present a result in which probabilistic methods are used to prove existence and uniqueness of a backward partial differential equation in a Hilbert space. This equation is of the form (7) in Theorem 1.1 below. In particular semi-linear conditions on the coefficient $f$ are imposed.
Citation: Jan A. Van Casteren. On backward stochastic differential equations in infinite dimensions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 803-824. doi: 10.3934/dcdss.2013.6.803
References:
[1]

V. Bally, E. Pardoux and L. Stoica, Backward stochastic differential equations associated to a symmetric Markov process, Potential Anal., 22 (2005), 17-60.

[2]

Felix E. Browder, Nonlinear elliptic boundary value problems, Bull. Amer. Math. Soc., 69 (1963), 862-874.

[3]

Felix E. Browder, Existence and perturbation theorems for nonlinear maximal monotone operators in Banach spaces, Bull. Amer. Math. Soc., 73 (1967), 322-327.

[4]

Fulvia Confortola, Dissipative backward stochastic differential equations in infinite dimensions, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 9 (2006), 155-168. doi: 10.1142/S0219025706002287.

[5]

Fulvia Confortola, Dissipative backward stochastic differential equations with locally Lipschitz nonlinearity, Stochastic Process. Appl., 117 (2007), 613-628. doi: 10.1016/j.spa.2006.09.008.

[6]

N. El Karoui and M. C. Quenez, Imperfect markets and backward stochastic differential equations, Numerical methods in finance, Publ. Newton Inst., Cambridge Univ. Press, Cambridge, (1997), 181-214.

[7]

Marco Fuhrman and Ying Hu, Backward stochastic differential equations in infinite dimensions with continuous driver and applications, Appl. Math. Optim., 56 (2007), 265-302. doi: 10.1007/s00245-007-0897-2.

[8]

Giuseppina Guatteri, On a class of forward-backward stochastic differential systems in infinite dimensions, J. Appl. Math. Stoch. Anal., (2007), Art. ID 42640, pp.33. doi: 10.1155/2007/42640.

[9]

Ying Hu and Shi Ge Peng, Adapted solution of a backward semilinear stochastic evolution equation, Stochastic Anal. Appl., 9 (1991), 445-459. doi: 10.1080/07362999108809250.

[10]

N. Ikeda and S. Watanabe, "Stochastic Differential Equations and Diffusion Processes," 2 ed., North-Holland Mathematical Library, 24, North-Holland, Amsterdam, 1998.

[11]

Antoine Lejay, BSDE driven by Dirichlet process and semi-linear parabolic PDE. Application to homogenization, Stochastic Process. Appl., 97 (2002), 1-39. doi: 10.1016/S0304-4149(01)00124-7.

[12]

George J. Minty, On a "Monotonicity'' method for the solution of non-linear equations in Banach spaces, Proc. Nat. Acad. Sci. U.S.A., 50 (1963), 1038-1041.

[13]

É. Pardoux, Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order, Stochastic analysis and related topics, VI (Geilo, 1996), Progr. Probab., 42, Birkhäuser Boston, Boston, MA, (1998), 79-127.

[14]

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

[15]

Étienne Pardoux and Aurel Răşcanu, Backward stochastic differential equations with subdifferential operator and related variational inequalities, (English summary) Stochastic Process. Appl., 76 (1998), 191-215. doi: 10.1016/S0304-4149(98)00030-1.

[16]

Étienne Pardoux and Aurel Răşcanu, Backward stochastic variational inequalities, Stochastics Stochastics Rep., 67 (1999), 159-167.

[17]

Étienne Pardoux and S. Zhang, Generalized BSDEs and nonlinear Neumann boundary value problems, Probab. Theory Related Fields, 110 (1998), 535-558. doi: 10.1007/s004400050158.

[18]

Jan Prüss, Maximal regularity for evolution equations in $L_p$-spaces, Conf. Semin. Mat. Univ. Bari, (2002), 1-39. (2003).

[19]

R. Tyrrell Rockafellar, Local boundedness of nonlinear, monotone operators, Michigan Math. J., 16 (1969), 397-407.

[20]

R. Tyrrell Rockafellar, "Convex Analysis," Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997, Reprint of the 1970 original, Princeton Paperbacks.

[21]

Elias M. Stein and Rami Shakarchi, "Real Analysis," Princeton Lectures in Analysis, III, Princeton University Press, Princeton, NJ, 2005, Measure theory, integration, and Hilbert spaces.

[22]

J. A. Van Casteren, Feynman-Kac formulas, backward stochastic differential equations and Markov processes, Functional Analysis and Evolution Equations, 83-111, Birkhäuser, Basel, 2008. doi: 10.1007/978-3-7643-7794-6_6.

[23]

J. A.Van Casteren, Viscosity solutions, backward stochastic differential equations and Mar\-kov processes, IMTA, Integration: Mathematical Theory and Applications, 1, no. 4 (2010), 273-420, Nova Publishers, Inc.

[24]

J. A. Van Casteren, "Markov Processes, Feller Semigroups and Evolution Equations,'' Series on Concrete and Applicable Mathematics 12, November 2010, WSPC, Singapore, London.

show all references

References:
[1]

V. Bally, E. Pardoux and L. Stoica, Backward stochastic differential equations associated to a symmetric Markov process, Potential Anal., 22 (2005), 17-60.

[2]

Felix E. Browder, Nonlinear elliptic boundary value problems, Bull. Amer. Math. Soc., 69 (1963), 862-874.

[3]

Felix E. Browder, Existence and perturbation theorems for nonlinear maximal monotone operators in Banach spaces, Bull. Amer. Math. Soc., 73 (1967), 322-327.

[4]

Fulvia Confortola, Dissipative backward stochastic differential equations in infinite dimensions, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 9 (2006), 155-168. doi: 10.1142/S0219025706002287.

[5]

Fulvia Confortola, Dissipative backward stochastic differential equations with locally Lipschitz nonlinearity, Stochastic Process. Appl., 117 (2007), 613-628. doi: 10.1016/j.spa.2006.09.008.

[6]

N. El Karoui and M. C. Quenez, Imperfect markets and backward stochastic differential equations, Numerical methods in finance, Publ. Newton Inst., Cambridge Univ. Press, Cambridge, (1997), 181-214.

[7]

Marco Fuhrman and Ying Hu, Backward stochastic differential equations in infinite dimensions with continuous driver and applications, Appl. Math. Optim., 56 (2007), 265-302. doi: 10.1007/s00245-007-0897-2.

[8]

Giuseppina Guatteri, On a class of forward-backward stochastic differential systems in infinite dimensions, J. Appl. Math. Stoch. Anal., (2007), Art. ID 42640, pp.33. doi: 10.1155/2007/42640.

[9]

Ying Hu and Shi Ge Peng, Adapted solution of a backward semilinear stochastic evolution equation, Stochastic Anal. Appl., 9 (1991), 445-459. doi: 10.1080/07362999108809250.

[10]

N. Ikeda and S. Watanabe, "Stochastic Differential Equations and Diffusion Processes," 2 ed., North-Holland Mathematical Library, 24, North-Holland, Amsterdam, 1998.

[11]

Antoine Lejay, BSDE driven by Dirichlet process and semi-linear parabolic PDE. Application to homogenization, Stochastic Process. Appl., 97 (2002), 1-39. doi: 10.1016/S0304-4149(01)00124-7.

[12]

George J. Minty, On a "Monotonicity'' method for the solution of non-linear equations in Banach spaces, Proc. Nat. Acad. Sci. U.S.A., 50 (1963), 1038-1041.

[13]

É. Pardoux, Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order, Stochastic analysis and related topics, VI (Geilo, 1996), Progr. Probab., 42, Birkhäuser Boston, Boston, MA, (1998), 79-127.

[14]

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

[15]

Étienne Pardoux and Aurel Răşcanu, Backward stochastic differential equations with subdifferential operator and related variational inequalities, (English summary) Stochastic Process. Appl., 76 (1998), 191-215. doi: 10.1016/S0304-4149(98)00030-1.

[16]

Étienne Pardoux and Aurel Răşcanu, Backward stochastic variational inequalities, Stochastics Stochastics Rep., 67 (1999), 159-167.

[17]

Étienne Pardoux and S. Zhang, Generalized BSDEs and nonlinear Neumann boundary value problems, Probab. Theory Related Fields, 110 (1998), 535-558. doi: 10.1007/s004400050158.

[18]

Jan Prüss, Maximal regularity for evolution equations in $L_p$-spaces, Conf. Semin. Mat. Univ. Bari, (2002), 1-39. (2003).

[19]

R. Tyrrell Rockafellar, Local boundedness of nonlinear, monotone operators, Michigan Math. J., 16 (1969), 397-407.

[20]

R. Tyrrell Rockafellar, "Convex Analysis," Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997, Reprint of the 1970 original, Princeton Paperbacks.

[21]

Elias M. Stein and Rami Shakarchi, "Real Analysis," Princeton Lectures in Analysis, III, Princeton University Press, Princeton, NJ, 2005, Measure theory, integration, and Hilbert spaces.

[22]

J. A. Van Casteren, Feynman-Kac formulas, backward stochastic differential equations and Markov processes, Functional Analysis and Evolution Equations, 83-111, Birkhäuser, Basel, 2008. doi: 10.1007/978-3-7643-7794-6_6.

[23]

J. A.Van Casteren, Viscosity solutions, backward stochastic differential equations and Mar\-kov processes, IMTA, Integration: Mathematical Theory and Applications, 1, no. 4 (2010), 273-420, Nova Publishers, Inc.

[24]

J. A. Van Casteren, "Markov Processes, Feller Semigroups and Evolution Equations,'' Series on Concrete and Applicable Mathematics 12, November 2010, WSPC, Singapore, London.

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