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November  2015, 35(11): 5285-5315. doi: 10.3934/dcds.2015.35.5285

## 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

Received  October 2013 Revised  October 2014 Published  May 2015

In this paper we study the solvability of backward doubly stochastic differential equations (BDSDEs for short) with polynomial growth coefficients and their connections with SPDEs. The corresponding SPDE is in a very general form, which may depend on the derivative of the solution. We use Wiener-Sobolev compactness arguments to derive a strongly convergent subsequence of approximating SPDEs. For this, we prove some new estimates to the solution and its Malliavin derivative of the corresponding approximating BDSDEs. These estimates lead to the verifications of the conditions in the Wiener-Sobolev compactness theorem and the solvability of the BDSDEs and the SPDEs with polynomial growth coefficients.
Citation: Qi Zhang, Huaizhong Zhao. Backward doubly stochastic differential equations with polynomial growth coefficients. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5285-5315. doi: 10.3934/dcds.2015.35.5285
##### 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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##### 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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