# 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.   Google Scholar [2] Felix E. Browder, Nonlinear elliptic boundary value problems,, Bull. Amer. Math. Soc., 69 (1963), 862.   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.   Google Scholar [4] Fulvia Confortola, Dissipative backward stochastic differential equations in infinite dimensions,, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 9 (2006), 155.  doi: 10.1142/S0219025706002287.  Google Scholar [5] Fulvia Confortola, Dissipative backward stochastic differential equations with locally Lipschitz nonlinearity,, Stochastic Process. Appl., 117 (2007), 613.  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, (1997), 181.   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.  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).  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.  doi: 10.1080/07362999108809250.  Google Scholar [10] N. Ikeda and S. Watanabe, "Stochastic Differential Equations and Diffusion Processes,", 2 ed., 24 (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.  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.   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, 42 (1998), 79.   Google Scholar [14] Étienne Pardoux and S. G. 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 [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.  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.   Google Scholar [17] Étienne Pardoux and S. Zhang, Generalized BSDEs and nonlinear Neumann boundary value problems,, Probab. Theory Related Fields, 110 (1998), 535.  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.   Google Scholar [19] R. Tyrrell Rockafellar, Local boundedness of nonlinear, monotone operators,, Michigan Math. J., 16 (1969), 397.   Google Scholar [20] R. Tyrrell Rockafellar, "Convex Analysis,", Princeton Landmarks in Mathematics, (1997).   Google Scholar [21] Elias M. Stein and Rami Shakarchi, "Real Analysis,", Princeton Lectures in Analysis, (2005).   Google Scholar [22] J. A. Van Casteren, Feynman-Kac formulas, backward stochastic differential equations and Markov processes,, Functional Analysis and Evolution Equations, (2008), 83.  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, 1 (2010), 273.   Google Scholar [24] J. A. Van Casteren, "Markov Processes, Feller Semigroups and Evolution Equations,'', Series on Concrete and Applicable Mathematics 12, 12 (2010).   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.   Google Scholar [2] Felix E. Browder, Nonlinear elliptic boundary value problems,, Bull. Amer. Math. Soc., 69 (1963), 862.   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.   Google Scholar [4] Fulvia Confortola, Dissipative backward stochastic differential equations in infinite dimensions,, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 9 (2006), 155.  doi: 10.1142/S0219025706002287.  Google Scholar [5] Fulvia Confortola, Dissipative backward stochastic differential equations with locally Lipschitz nonlinearity,, Stochastic Process. Appl., 117 (2007), 613.  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, (1997), 181.   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.  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).  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.  doi: 10.1080/07362999108809250.  Google Scholar [10] N. Ikeda and S. Watanabe, "Stochastic Differential Equations and Diffusion Processes,", 2 ed., 24 (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.  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.   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, 42 (1998), 79.   Google Scholar [14] Étienne Pardoux and S. G. 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 [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.  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.   Google Scholar [17] Étienne Pardoux and S. Zhang, Generalized BSDEs and nonlinear Neumann boundary value problems,, Probab. Theory Related Fields, 110 (1998), 535.  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.   Google Scholar [19] R. Tyrrell Rockafellar, Local boundedness of nonlinear, monotone operators,, Michigan Math. J., 16 (1969), 397.   Google Scholar [20] R. Tyrrell Rockafellar, "Convex Analysis,", Princeton Landmarks in Mathematics, (1997).   Google Scholar [21] Elias M. Stein and Rami Shakarchi, "Real Analysis,", Princeton Lectures in Analysis, (2005).   Google Scholar [22] J. A. Van Casteren, Feynman-Kac formulas, backward stochastic differential equations and Markov processes,, Functional Analysis and Evolution Equations, (2008), 83.  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, 1 (2010), 273.   Google Scholar [24] J. A. Van Casteren, "Markov Processes, Feller Semigroups and Evolution Equations,'', Series on Concrete and Applicable Mathematics 12, 12 (2010).   Google Scholar
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