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On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions: The critical case
Backward doubly stochastic differential equations with polynomial growth coefficients
1. | School of Mathematical Sciences, Fudan University, Shanghai 200433 |
2. | Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, United Kingdom |
References:
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V. Bally and B. Saussereau, A relative compactness criterion in Wiener-Sobolev spaces and application to semi-linear stochastic PDEs, J. Funct. Anal., 210 (2004), 465-515.
doi: 10.1016/S0022-1236(03)00236-2. |
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C. R. Feng and H. Z. Zhao, Random periodic solutions of SPDEs via integral equations and Wiener-Sobolev compact embedding, J. Funct. Anal., 262 (2012), 4377-4422.
doi: 10.1016/j.jfa.2012.02.024. |
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D. Nualart, The Malliavin Calculus and Related Topics, Springer-Verlag, New York, 1995.
doi: 10.1007/978-1-4757-2437-0. |
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E. Pardoux, BSDE's weak convergence and homogenization of semilinear PDE's, in Nonlinear Analysis, Differential Equations and Control (eds. F. Clarke and R. Stern), Kluwer Acad. Publi., 528, Dordrecht, 1999, 503-549. |
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E. Pardoux and S. Peng, Backward doubly stochastic differential equations and systems of quasilinear SPDEs, Probab. Theory Rel., 98 (1994), 209-227.
doi: 10.1007/BF01192514. |
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S. Peszat, On a Sobolev space of functions of infinite number of variables, Bull. Polish Acad. Sci. Math., 41 (1993), 55-60. |
[8] |
Q. Zhang and H. Z. Zhao, Stationary solutions of SPDEs and infinite horizon BDSDEs, J. Funct. Anal., 252 (2007), 171-219.
doi: 10.1016/j.jfa.2007.06.019. |
[9] |
Q. Zhang and H. Z. Zhao, Stationary solutions of SPDEs and infinite horizon BDSDEs with non-Lipschitz coefficients, J. Differ. Equations, 248 (2010), 953-991.
doi: 10.1016/j.jde.2009.12.013. |
[10] |
Q. Zhang and H. Z. Zhao, Probabilistic representation of weak solutions of partial differential equations with polynomial growth coefficients, J. Theor. Probab., 25 (2012), 396-423.
doi: 10.1007/s10959-011-0350-y. |
[11] |
Q. Zhang and H. Z. Zhao, SPDEs with polynomial growth coefficients and Malliavin calculus method, Stoch. Proc. Appl., 123 (2013), 2228-2271.
doi: 10.1016/j.spa.2013.02.004. |
show all references
References:
[1] |
V. Bally and B. Saussereau, A relative compactness criterion in Wiener-Sobolev spaces and application to semi-linear stochastic PDEs, J. Funct. Anal., 210 (2004), 465-515.
doi: 10.1016/S0022-1236(03)00236-2. |
[2] |
G. Da Prato, P. Malliavin and D. Nualart, Compact families of Wiener functionals, C. R. Acad. Sci. Paris Ser. I Math., 315 (1992), 1287-1291. |
[3] |
C. R. Feng and H. Z. Zhao, Random periodic solutions of SPDEs via integral equations and Wiener-Sobolev compact embedding, J. Funct. Anal., 262 (2012), 4377-4422.
doi: 10.1016/j.jfa.2012.02.024. |
[4] |
D. Nualart, The Malliavin Calculus and Related Topics, Springer-Verlag, New York, 1995.
doi: 10.1007/978-1-4757-2437-0. |
[5] |
E. Pardoux, BSDE's weak convergence and homogenization of semilinear PDE's, in Nonlinear Analysis, Differential Equations and Control (eds. F. Clarke and R. Stern), Kluwer Acad. Publi., 528, Dordrecht, 1999, 503-549. |
[6] |
E. Pardoux and S. Peng, Backward doubly stochastic differential equations and systems of quasilinear SPDEs, Probab. Theory Rel., 98 (1994), 209-227.
doi: 10.1007/BF01192514. |
[7] |
S. Peszat, On a Sobolev space of functions of infinite number of variables, Bull. Polish Acad. Sci. Math., 41 (1993), 55-60. |
[8] |
Q. Zhang and H. Z. Zhao, Stationary solutions of SPDEs and infinite horizon BDSDEs, J. Funct. Anal., 252 (2007), 171-219.
doi: 10.1016/j.jfa.2007.06.019. |
[9] |
Q. Zhang and H. Z. Zhao, Stationary solutions of SPDEs and infinite horizon BDSDEs with non-Lipschitz coefficients, J. Differ. Equations, 248 (2010), 953-991.
doi: 10.1016/j.jde.2009.12.013. |
[10] |
Q. Zhang and H. Z. Zhao, Probabilistic representation of weak solutions of partial differential equations with polynomial growth coefficients, J. Theor. Probab., 25 (2012), 396-423.
doi: 10.1007/s10959-011-0350-y. |
[11] |
Q. Zhang and H. Z. Zhao, SPDEs with polynomial growth coefficients and Malliavin calculus method, Stoch. Proc. Appl., 123 (2013), 2228-2271.
doi: 10.1016/j.spa.2013.02.004. |
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