# American Institute of Mathematical Sciences

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 & 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.  Google Scholar [2] Felix E. Browder, Nonlinear elliptic boundary value problems, Bull. Amer. Math. Soc., 69 (1963), 862-874.  Google Scholar [3] Felix E. Browder, Existence and perturbation theorems for nonlinear maximal monotone operators in Banach spaces, Bull. Amer. Math. Soc., 73 (1967), 322-327.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] N. Ikeda and S. Watanabe, "Stochastic Differential Equations and Diffusion Processes," 2 ed., North-Holland Mathematical Library, 24, North-Holland, Amsterdam, 1998.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] Étienne Pardoux and Aurel Răşcanu, Backward stochastic variational inequalities, Stochastics Stochastics Rep., 67 (1999), 159-167.  Google Scholar [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.  Google Scholar [18] Jan Prüss, Maximal regularity for evolution equations in $L_p$-spaces, Conf. Semin. Mat. Univ. Bari, (2002), 1-39. (2003).  Google Scholar [19] R. Tyrrell Rockafellar, Local boundedness of nonlinear, monotone operators, Michigan Math. J., 16 (1969), 397-407.  Google Scholar [20] R. Tyrrell Rockafellar, "Convex Analysis," Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997, Reprint of the 1970 original, Princeton Paperbacks.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [24] J. A. Van Casteren, "Markov Processes, Feller Semigroups and Evolution Equations,'' Series on Concrete and Applicable Mathematics 12, November 2010, WSPC, Singapore, London. Google Scholar

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##### 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.  Google Scholar [2] Felix E. Browder, Nonlinear elliptic boundary value problems, Bull. Amer. Math. Soc., 69 (1963), 862-874.  Google Scholar [3] Felix E. Browder, Existence and perturbation theorems for nonlinear maximal monotone operators in Banach spaces, Bull. Amer. Math. Soc., 73 (1967), 322-327.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] N. Ikeda and S. Watanabe, "Stochastic Differential Equations and Diffusion Processes," 2 ed., North-Holland Mathematical Library, 24, North-Holland, Amsterdam, 1998.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] Étienne Pardoux and Aurel Răşcanu, Backward stochastic variational inequalities, Stochastics Stochastics Rep., 67 (1999), 159-167.  Google Scholar [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.  Google Scholar [18] Jan Prüss, Maximal regularity for evolution equations in $L_p$-spaces, Conf. Semin. Mat. Univ. Bari, (2002), 1-39. (2003).  Google Scholar [19] R. Tyrrell Rockafellar, Local boundedness of nonlinear, monotone operators, Michigan Math. J., 16 (1969), 397-407.  Google Scholar [20] R. Tyrrell Rockafellar, "Convex Analysis," Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997, Reprint of the 1970 original, Princeton Paperbacks.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [24] J. A. Van Casteren, "Markov Processes, Feller Semigroups and Evolution Equations,'' Series on Concrete and Applicable Mathematics 12, November 2010, WSPC, Singapore, London. Google Scholar
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